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: 
No, natively WordPress does not log any activity. While plugins for it exist, they won't work retroactively.

However don't forget that any action taken in WordPress is technically a HTTP request, such as page load, form submit, or Ajax request.

These are often logged pretty thoroughly in web server's access log and if available it's pretty realistic to reconstruct action that happened and details like user's IP from them.
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.