The good news is that playing piano with a healthy technique will not exacerbate tendonitis, but be likely to help it heal. However, explaining how to work to make your playing technique healthier is hard to do in only a few words, especially without knowing what types of tensions you currently have in your technique.
In short, I would say, work on being able to hold the keys firmly down in the keybed while keeping all joints of the shoulders, arm, hand, and fingers loose. (playing with dead spaghetti fingers that don't pull the keys to the bottom will not work).
A good but very very tricky exercise for developing this is: press down a cluster of notes, for example C D E F G using fingers 1 2 3 4 5 with each hand. While holding all these keys down and not allowing the to rise, and maintaining the fingertips' original position on the key, move your wrists around, up and down and left to right, in an as large circle as possible, while trying to make it feel as easy and effortless as possible. If this is too hard, start out with holding down anything between a second and a seventh using either fingers 12, 13, 14, or 15, and try to move your wrist up and down as much as possible while making it feel easy and effortless. You'll have to relax your thumb as well as your entire arm to be successfull at this. Also the shoulder will make this difficult if you tense it.