Poly, BCV and VWL and I think BWV have kids clubs in the evening, but children have to be 4 years old and potty trained.
You would need in room sitting. Several companies offer it in the Orlando area. We use Fairy Godmothers, and have always been very pleased. In fact, we'll probably continue to use them even though DS is now 4, so he can get to bed at a decent hour.
Disney recommends Kids Nite Out. You can check out their Web sites at kidsniteout.com.
Be warned there are lots of posters opposed to baby sitters, so some of them might lash out at you. But we use sitters all the time and have great luck with them.