Ambergris Caye is largely undeveloped with only one small town, and as such has no facilities designed to host the performing arts.  The closest thing you can find without leaving the island is to see some of the local musicians that serve as the evening entertainment at the bars and restaurants in San Pedro town.

However, if you are interested in something more sophisticated, you can take a trip into Belize City (20 minutes by plane, or 90 minutes by ferry).  There you will find the Bliss Institute for the Performing Arts.  Regular performances include theatrical, musical and dance productions as well as film festivals and other community and national events.  A collection of Belizean artwork is also on exhibit here.