A lot depends what level of exercise your hands and wrists can tolerate. Over time, it's likely moderate exercise will help for most folks, but it's hard to say for your case, specifically. You probably don't want anything with lots of wrist locking early on, nor a lot of grip fighting. But if it starts out relatively gentle, then you can gauge how it's affecting you. I'd expect a BJJ school to be able to accommodate that for a month or so while you find out how your hands react. You could do the same with some of the standing grappling styles (Aikido, Hapkido, Nihon Goshin Aikido, some Japanese Jujutsu). Toss Judo in there as an option to toy with, though both it and BJJ will eventually need you to be able to use your grip with strength on a regular basis. Pretty much any grappling will need that, but I think it's more a need/focus with styles that compete, especially if they compete in gi.
For me, striking has less of an aggravating effect on my tendinitis, but that's probably because of where it is. For some people with tenderness in the wrists, punching will just make it worse. If you try out striking, I'd ask the instructor to let you try some soft, slow drills on a target (heavy bag, whatever) under supervision, so you can see how your hands and wrists react when you do it softly and "correctly".
