I'm not sure about how to find work in the islands, but I know that a lot of the Carribean countries have pretty strict work permit requirements. I understand it can be difficult to get permits, even for citizens of a country of which an island is a former colony (e.g. Britain and Bermuda).
You might want to check this out carefully, though as a U.S. citizen I think you'd be O.K. in Puerto Rico or the U.S. Virgin Islands.