Is there any way to allow the user to click a button on a Facebook canvas app and allow the app to go full-screen?
If it's possible, is it permitted by Facebook?
The web has a "request fullscreen" API that you can use (be sure to read up on in the reference section below). Facebook seems to indicate this is allowed to be use, just take a look at this transcript:
Then, some time later...
Simply retrieve your canvas element and call requestFullScreen: