Of course, this is hard to say over the Internet, but tendonitis would indeed be your first bet, but there could be a lot of things going on. Obviously, a correct diagnosis requires hand specialists to take a closer look at you.
If you are aggravating your wrist problems through playing the piano you may be playing with thumb orientation, i.e ulnar deviation, which in turn often comes form the wrong notion that fingers always need to be aligned with the forearm when they are playing. This principle applied to the thumb leads to ulnar deviation.
The bottom line is, you need to carefully analyze your technique, or better yet, have it analyzed by somebody who is knowledgeable about these things (e.g. a Taubman teacher). Don't you have knowledgeable people at your college? Given that most pianists are injured, there will be plenty of people around to talk to.
Your technique must be in line with your anatomy. This is true for all activities, from opening doors to pressing buttons. If your use of your body is correct, you will not aggravate any problems you may already have. Those can then be treated specifically, provided they are diagnosed correctly. You may have to stop playing the piano for a while to get that taken care of.