Yes Mate, theres loads of tennis courts. Since Paradorns success lately, it is really popular with the locals.
All the major hotels in Patong have courts......the best ones being The Patong Merlin, Diamond Cliffs, Patong Resort, Club Andaman.
A little farther afield but very nice is the indoor court at Laguna.
Phuket Country Club has tennis courts (a little scruffy) and near the jail in Phuket Town there are a whole heap of them. Not sure about how non-members would get on but I'm sure someone will be able to advise.
A friend of mine who I could put you in touch with, plays every day at "Phuket Land and House" and his son is in the Thai national team. I'm sure he could find you a game.