Short Answer: No.
Long answer: My wife is a physican and goes the Belize twice a year for secular medical mission work. She is pretty much the doctor for an entire region. The reefs there are protected. Sure - you could snag something but it's doubtful you'f get it back through customs. Let alone get it back alive. I don't know of any fish stores there - but again you probably couldn't get anything back to the states. Last time she went, she picked up some small pieces of broken coral that had washed up on the beach and brought those back. She put them in the checked bag. Had they been found they would have been confiscated and she would have been in trouble as you can't prove that you "found" them and didn't break them off the reef. I don't think she will do that again.