Question: Someone moved a very important page on my website into the trash, and I do not know who did it! It was not deleted permanently so I don't need to worry, in that sense.

The revisions, when I restored it, show that someone edited it 3 days prior to today, so it could have been them, but I can't be sure.

Does WP keep track of who clicks the trash button? If not, I presume I'd have to write a custom script to hook onto the trash button, when clicked.

Answer: 
WP can not do it, but you can go to the server logs and find the user's IP. It may take some effort and success is not guaranteed but this is the only possible way. Will's answer will help you in future mistakes.

Finding a user based on their IP is not usually hard especially if the number of authors is limited.
There is a free plugin called *Stream* that will make your life much easier. Search within the WordPress plugins page. It logs all changes to posts and who is responsible for them.