I recently purchased a 16ga Sterlingworth. I requested and received a factory letter for it. It was made in 1920 and shipped to a dealer in Havana. So, there was some form of hunting going on.

Our Canadian neighbors already have access.
http://www.gocuba.ca/en/hunting_fishing.asp
http://www.cubafun.com/english/caza.asp

Considering that Cuba was once a Spanish possession, I am sure that there must be a long history of hunting there.

Pete