It’s impossible to prove a negative, but I’m rather sure there is nothing official that covers this. To that end, here are all the most likely sources across all of D&D to cover something like this, and how they don’t.
Note that numerous sources discuss how fiends all have names other than their “true name,” that they could use for this purpose. As discussed at the end of the answer, real-world contracts wouldn’t require the equivalent of a “true name,” and can even be entered without any name at all. Notably, the recent Baldur’s Gate: Descent into Avernus adventure included several forms of infernal bargains, most of which don’t involve anyone signing anything.
Original D&D and other early editions
Things were not as detailed this far back and a lot of it has been ret-conned since. There may be something about infernal bargains somewhere in the corpus, but finding it may prove beyond my ability.
This far back you also get a whole smorgasbord of different versions of the game, some of them by different authors, not all of them neatly numbered, as TSR sought to flesh out the original rules even as it moved on to the “Advanced” rules. That’s where you get BECMI and the like, the Holmes and Moldvay versions of the game. BECMI in particular was set in Mystara, which is sort of wildly (and intentionally) incompatible with the general cosmology of D&D, so even if it covered things it probably wouldn’t apply to the rest of D&D.
AD&D is, as I understand things, kind of where D&D began to get more cohesive, and that’s where we start to see setting details like this develop more. We still don’t get any information on exactly how infernal bargains are signed on the devil side.
My thanks to @Omortis for this information, since I do not have access to any library of AD&D books.
Monster Manual, as in, the first one
Devils are discussed, but no details on how devils extract value (e.g. souls) from deals with mortals are offered—instead, it focuses on how a mortal can convince or force a devil to do their bidding.
Fiend Folio was like Monster Manual, only more evil (but not exclusively so, much less exclusively about fiends). Anyway, no mention of infernal bargains is found here.
Monster Manual II
Having apparently exhausted their thesaurus with Fiend Folio, AD&D goes back to Monster Manual a second time. Now, we get the tiniest hint of making deals with devils, but it’s rather scant:
[Getting a devil to do something for you] also typically requires a contract for the soul of the creature commanding the infernal power to obey
(Quotation supplied by Omortis, citing Monster Manual II)
and that’s it.
AD&D 2e is where the cosmology of D&D really develops—this is where it got its own product line as the Planescape setting. It also has a ton of material. I have spoken with Planescape expert @afroakuma on this, since like the original books and the first edition of AD&D, I don’t have a 2e library.
Faces of Evil: The Fiends
My understanding is that Faces of Evil: The Fiends is the 2e book most likely to cover this. However, I’ve confirmed it does not. It talks about true names and how important it is to keep them secret, and it mentions how important corrupting mortals is to the devils, but it doesn’t get into the details of how infernal bargains are actually made, much less how they’re signed.
A Guide to Hell
I want to also look through A Guide to Hell, if I can only find a copy—so far, I have struck out on that.
D&D 3e & 3.5e
The third edition, and “v.3.5 revised edition,” of D&D had a number of supplements devoted to devils specifically or fiends more generally. None of them cover this point.
Book of Vile Darkness
Book of Vile Darkness is a weird, “3.25” book published just before the “v.3.5 revised edition” and so it reflects some but not all of the updates from that revision. It also had a prominent sticker on it warning it was for “mature audiences only,” though honestly most of the content is pretty standard early-2000s “edgy teen” schlock.
Anyway, one of the topics covered in the book is “Souls as Power,” which spends a few paragraphs discussing the idea that you can use a soul, once you have it, to power up magic—there are rules for using souls as material components to power-up spells, or using souls instead of one’s own XP when crafting magic items. (3.x had magic items cost XP to create; it was an awkward system.) Aaaand—that’s it. There is a little bit about how one can harvest souls from the Lower Realms, but not about how you corrupt mortals so their souls go there in the first place.
Otherwise, the book contains a bunch of new evil prestige classes (a 3.x term, a class that you could only multiclass into after meeting certain requirements), new evil spells, new evil magic items, and new evil creatures. Scanning the lists of each, none of them touch on dealing for souls. There’s soul shackles, a spell that can be cast on a creature already dead to imprison its soul and allow the caster to interrogate it, but that’s as close as we get. The only other vaguely-related option I see is a souldrinking weapon, which curses targets and grants the wielder benefits for doing so.
Wizards of the Coast reused the Fiend Folio title for another “3.25-ish” book, and again it’s kind of like a more evil Monster Manual, though honestly its collection of actual fiends is pretty limited. There are two demons (sentient evil pollution and a vampiric fiend), two devils (a spymaster and a thought-stealer), and two yugoloths (a sergeant in their armies and a spy). None of these descriptions get into signing away one’s soul.
However, towards the back of the book, Fiend Folio has a few fiend-specific prestige classes, and most relevantly here, the fiend of corruption prestige class. Fiend of corruption is all about making deals for souls, and gets a number of abilities to empower mortals who sign on the dotted line (ability score empowerment, free wealth via major creation, and ultimately, whatever they want via wish). The art for the class features a harrowed-looking man with a pen over parchment, as a nude lady-fiend caresses him and wraps him in her wings.
But as to actually signing away one’s soul, all the class offers is
Soul Bargain (Su): At 6th level, a fiend of corruption gains the ability to enter a binding agreement with a mortal, at the cost of the mortal’s soul. The mortal victim must enter into the soul bargain willingly.
Upon the mortal’s death (by any means), her soul is transferred to a gem (prepared as with the soul bind spell when the bargain is forged), even if the gem and the mortal are not on the same plane at the time.
The bargain requires 1 hour to complete, and is utterly inviolable once forged. The only way to escape a soul bargain is to recover the gem after the mortal’s death and break it, freeing her soul and allowing her to be restored to life through the normal means.
It is quite common for fiends of corruption, as soon as a soul bargain is complete, to return to their native plane and await the mortal’s death, or at least deposit the gem in a safe place before returning to the Material Plane to hasten the victim’s demise.
(Fiend Folio pg. 204)
And that’s it, the entirety of the ability. It does reference the spell soul bind, but the rules for that “gem (prepared as with the soul bind spell” are merely
Focus: A black sapphire of at least 1,000 gp value for every Hit Die possessed by the creature whose soul is to be bound. If the gem is not valuable enough, it shatters when the binding is attempted. (While creatures have no concept of level or Hit Dice as such, the value of the gem needed to trap an individual can be researched. Remember that this value can change over time as creatures gain more Hit Dice.)
(Player’s Handbook pg. 281)
So that doesn’t tell us anything about signing anything away.
Fiendish Codex II: Tyrants of the Nine Hells
Following up Fiendish Codex I: Hordes of the Abyss, which was about Chaotic Evil demons who don’t really bother with contracts, Fiendish Codex II is all about Lawful Evil devils, very much love contracts. The book discusses the “Pact Primeval” from which Asmodeus and the baatezu get their power, and various other “Faustian pacts” that devils can offer to mortals. The book devotes a few pages to this, but never mentions anything about how the pact is signed, on either side.
Heroes of Horror
Heroes of Horror has a much more Frankenstein, Dr. Jekyll and Mr. Hyde kind of aesthetic, with a bit of Lovecraft thrown in, and so doesn’t focus so much on actual devils. It even refers to fiends as “obvious” and “familiar” sources of evil villains, suggesting that they are thus less suitable for “horror,” and pushes a DM to maybe try out some less-used outsiders like genies and inevitables rather than fiends. And that’s the entirety of their section on “Outsiders as Villains of Horror.”
The only 4e book to touch significantly on devils is The Plane Above, weirdly enough, since The Plane Below is the Elemental Chaos and devils avoid chaos. Anyway, neither book discusses infernal contracts in any fashion.
There’s really only one book in 5e to look to here, and it doesn’t provide an answer either.
Baldur’s Gate: Descent into Avernus
Descent into Avernus is primarily an adventure, but it does describe several possible benefits to selling one’s soul. It does not, however, discuss how that gets signed off on by the fiend on the other side, though there is mention of that happening:
To enter the contract, the devil and the character making the deal sign the scroll in their own blood,
(Baldur’s Gate: Descent into Avernus pg. 215)
But more importantly, that description applies only to one of six options for the format of an infernal contract:
||Song of the devil
||Written in stone
(Baldur’s Gate: Descent into Avernus pg. 214)
This clearly demonstrates that a deal with a devil doesn’t necessarily involve a signature at all. The infernal scroll does, but the others don’t. (Written in stone, for the curious, has each “sign” the stone by pressing their hand into it.)
Per the above, I believe there is no official answer to this question. I would therefore like to address a misunderstanding in it, namely that
surely [devils] need to put their full names somewhere on the contract to be binding?
This is untrue.
In real life, there are lots of ways to sign contracts, both in modern times and historically. Devils aren’t the only ones who want to hide their true identity when entering into a contract, you know.
The most likely form of signature for a devil lord isn’t a “signature” at all, but a seal. Seals were—and still are—used to indicate that someone in authority has agreed to something, with greater security and gravitas than a signature can manage. They are harder to forge, much harder to modify after the fact, and also can have the side-benefit of declaring the greatness of whoever’s seal it is to anyone who sees it. This is extremely fitting to devils, who after all tend to style themselves as “dukes” and “archdukes.”
Common among the illiterate, any kind of mark on the page that is reasonably unique can be used in lieu of actual letters. When it really wasn’t that important, a simple “X” has even been used. In the case of an archduke of Hell, you’d expect something fancier than this, but then, especially since they have magic available to assist, perhaps their mark could be very fancy indeed—more like a seal, but without the opportunity to steal it.
Any other identifying term
Real-world contracts do not require one’s full legal name—they require enough information to uniquely identify an individual. I live in a city with about a dozen people who share my entire name, for example—my full name alone does not identify me, which is why my mortgage has a bunch of other information about me to make sure we know which Kevin Ryan we’re talking about. When it comes time to enforce a contract, the court is going to read it and if the signatory wants to argue “that’s not me,” they’ll look into how the person is identified and whether or not it suitably establishes that it really was.
Considering that the infernal scroll is signed in the blood of both the devil and the mortal, magic probably can trace that to each of them and it doesn’t matter what they write on it. It can be a fake name, nickname, a title, whatever, it doesn’t matter. We know from numerous sources that outsiders have “given names” that they hand out to people for various purposes, to avoid using their “true name,” so they probably mostly use that, but it could be almost anything—with their blood there to identify them, the actual text makes no difference.
This is how real-world individuals and businesses enter into contracts that they don’t want associated with their own names. Devils can do it too.