The only club in the area is the Sport Shooting club and, as far as I know they don't do any clay shooting. To become a member of the club you have to be introduced by an existing member.
It might be worth your while going to speak to Mehmet who owns the gun shop at the Fish Market. He speaks reasonable English and may be able to get you an introduction to somebody who knows where you might be able to go clay shooting.
Just a word of warning though. I've been out hunting with locals and, to be quite frank, I wouldn't go again. Gun safety doesn't seem to be in their vocabulary.