Quite a few options. There are Disney slow-pitch softball leagues, a sand volleyball league (both start in mid February, generally), and a walking league (you earn points for your team based on how far you walk). There are also soccer and 3-3 basketball tournaments, all of which are sponsored by the Cast Activities department.
Outside of that, I believe the apartment complexes have tennis courts, and if not, then there are tennis courts at Mickey's Retreat, which is a cast member recreation area within walking distance of Chatham, Patterson, and The Commons. Mickey's retreat is where the soccer and volleyball leagues/tournaments are held, and a lot of folks get together for pick-up soccer games.
For the leagues, a lot (not all) of the teams are made up of people that work together. Ask around to see if your area has a team ... if not, you can get in touch with the Cast Activities office, and they'll pass your info on to the captains ... From experience, some teams never have enough players, so it's not hard to get picked up. (I play softball and volleyball ... as of right now, there are no open spots on either team, but that could change)