There are affiliate software that can track affiliates via other means, e.g. session and/or cookies. It depends on the platform you use. But search in hotscripts.com, there are many solutions/scripts available.

If your 'Thank you' page is actually on someone else's site which you are not allowed to put any tracking code on, your quite stuck I'm afraid.

You could try some hackish stuff like loading their page in a frame and checking if the loaded URL is the 'thank you' page, but it doesn't sound like a great idea.