As much as I like having the hardcover core rule-books in front of me, usually I read or reference most information for RPGs on my computer. Unfortunately, my google-fu has failed me at finding places to buy a Player's Handbook PDF, or even whether said PDF exists at all. My question is this: Is there any place I can go to legally buy a D&D 5e core rulebook PDF? I'm not looking for the starter set or basic rules, since those are incomplete (and free), I'm just looking for the core books in PDF format.
Not as of October 2017.
Per the Mike Mearls interview of 3 November 2014:
We're definitely looking at PDFs, ebooks, and other digital platforms, but no news yet. The goal with anything along those lines will be grow D&D, not just sell ebooks to people who already play the game, so we're putting a lot of work into figuring out that side of the equation.
It is likely that if and when a PDF version is made available, it will appear on http://www.dndclassics.com/ alongside the other PDF material from Wizards of the Coast.
January 2016 Update: The 5E system reference document is available in PDF format. For many people this may fulfill the need for reference material (looking up spells and creatures, etc)
October 2017 Update: While a PDF version is still not available, there are now a number of options for electronic reference.
There is one possibility, though I doubt you'll like it.
Assuming US jurisdiction, I believe it's legal for you to create your own PDF copy of the rules, working from your own legally owned dead-tree copy of the rules. Of course it's a lot of typing, and you're not likely to want to go through the trouble. (And I can't say I'd blame you for that.) Just make sure you don't distribute said copy to anyone else, or you'll be violating copyright laws, and the penalties for that can be pretty freaking steep.
No, the 5e books are not officially available in electronic form, besides the free Basic rules and the partial rules placed in the SRD. If they ever do appear for sale, it'll likely be on the DMs Guild Web store.
The one option available to you is to scan the books into electronic form yourself, or have it done for you. While I am not a lawyer, here is the relevant information for you to make your own determination.
Though in the US it's not yet settled case law as to whether this is absolutely legal or not, there is a lot of reason to believe it is. Earlier court rulings on music and DVRs are only partially relevant; there is very recent legal precedent, however, specific to creating electronic versions of books. The Author's Guild generally calls any transformation of a published work into electronic form copyright infringement and would say you can't do this. However as recently as June of 2014, they suffered severe setbacks in Authors Guild v. Google and Authors Guild v. Hathitrust which significantly solidified the legal definition of fair use for published works. There is even a popular US service that has been in business since 2011 called 1dollarscan.com which will take in books from you and scan them (destroying the physical copy and providing you electronic copies in various formats) for your personal use. It has weathered all its legal challenges so far. (This is actually a popular practice imported from Japan where it's called "jisui," and it had back and forth legal challenges there too, ending with the scanning shops being declared illegal in Sept. 2013.) Until this exact use case comes before a court it is not possible to say it is "absolutely" legal, and it's going to vary by locality, but I think it's quite reasonable to conclude this falls under fair use as currently interpreted by the courts.
The rest of this answer is now outdated, due to DungeonScape closing down.
However, you should be able to get the equivalent eventually. DungeonScape is the online companion to D&D 5e, and should be available in the coming weeks or months.
Notably, this blog post talks about digital distribution, and mentions
In DungeonScape, taking advantage of the full suite of player-based tools (i.e. character creation and character sheet) requires the digital version of the PHB (Player’s Handbook) to be unlocked for each account/player.
This suggests that there will definitely be a way to acquire the digital PHB, if not PDF.
There is now a perfectly legal way of obtaining all the 5e data in a digital form. It is a program called Fantasy Grounds and it supports many other RPG's as well, it allows for conducting your games on-line, and has a free demo so you can try it out to see if it will be fit for your purpose. Also it is available on Steam in various package forms. Hopefully this will lead anyone who comes here looking for the undeniably legal way of acquiring this content to the best option at present.
(comment of 8 Dec 2016, by Bloodcinder): Fantasy Grounds does indeed include all the product identity material in the PHB, even the material that is not included in the SRD.
While it's unlikely that there will be PDF versions of the rulebooks available, as jbowtie's answer suggests, there is official confirmation that the complete official rules will be available on D&D Beyond, the 5th Edition digital compendium. BadEye, the Product Lead for D&D Beyond, said this:
We will absolutely have all the official content available as we get a little further down the road - thanks again for helping us test!
So it seems that, once Beyond leaves the Beta stage, it will have a full compendium of all of the 5th Edition D&D rules. At the moment, only the freely available Basic Rules are on Beyond.
In Denmark it is legal for you to create a digital backup of a product you have legally bought. As far as I know, this includes downloading it from the Internet, as long as you don't distribute it. I don't know if it works like this in other countries, but it might be legal for you to torrent it (disable upload first), as long as you have bought the physical book.
There are services that will scan any book you own into a searchable pdf word document such as bound book scanning. They have 2 versions, 1 destructive 2 non destructive. I priced it out and it's about 50$ to 80$ for non destructive depending on the options you get. Destructive is cheaper (32$ to 67).
I don't think it's worth it, but that's just me.