There aren’t any explicit rules that give proficiency or advantage for filling out forms correctly, but there are many traits and character options that could reasonably be considered to help.
Consider what is included as part of a broad skill proficiency given your game world and the character’s specific backgrounds
While the forms are of a simple “yes/no” type, understanding what the denizens of Mechanus are asking, and their expectations, involves knowledge of the planes. So I think there’s a case for Intelligence (Arcana) to apply to the roll, or possibly Intelligence (Religion) if the religion includes such knowledge.
Religion might also apply if the character is a member of a church which was quite officious - keeping meticulous parish records, writing indulgences, hand copying manuscripts etc. That could apply to a cleric or a monk.
One tool proficiency might apply: Calligrapher’s supplies
Someone proficient in calligraphy clearly has skill with a variety of writing implements, and as it’s not something that would otherwise come up often, I’d be inclined to allow it to apply. At the very least they’re less likely to make the equivalent of “typos” when filling out the form.
Proficiency with a forgery kit could apply, but I’d only rule that it did if the characters are lying on the forms or trying to copy answers from another source. Even then…maybe not.
Some backgrounds might suggest experience with filling out forms
An Acolyte, Guild Artisan or Merchant, Noble or Sage is very likely to have done a lot of paperwork. While there’s no specific proficiency involved, these backgrounds might suggest a skill they have would include relevant experience, or might justify advantage on a roll.
Other maybe suitable backgrounds could include Charlatan, Entertainer, Sailor, Soldier or Spy. And there are some from beyond the Player’s Handbook that might fit as well, including Archaeologist, City Investigator, Cloistered Scholar, Courtier or many of those available for Strixhaven students or the various Ravnica guilds, especially Azorius Functionary. (Thanks to Kirt for suggesting a few more.)
Order of Scribes Wizards might have an advantage
The Order of Scribes is an Arcane Tradition introduced in Tasha’s Cauldron of Everything. One of its key features is the “Wizardly Quill”, whose noted game effects are that it doesn’t require ink, allows the wizard to write much faster (at least when copying spell scrolls into their spell book), and can magically erase anything it’s written (allowing the wizard to correct mistakes without having to start over). While there aren’t any other concrete effects of the feature, this seems useful enough that I’d be inclined to offer advantage to such a wizard when filling out forms under pressure.
Some game effects can apply a benefit to any ability check
There are many of these, many applying to or supplied by Bards. They include Bardic Inspiration, non-bardic Inspiration, the Bard’s “Jack of All Trades” feature, and the guidance spell.
