Sharon, believe it or not, I am an animal lover and would never do that. Besides, anyone who can't tell a dogs eye from his ass shouldn't even have a gun.
My way would be to charge the neighbor with harboring a nuisance or disturbing the peace.

Another thing you might try, record his dog's yapping, find out when they sleep, call him up and play the recording of the yapping.
Personally, I would be most apt to confront them. You have a God given right to enjoy the peace and quiet of your home, and these idiots are robbing you of that.