Well, you can definitely hunt small game with an airgun and a small game license. There doesn't seem to be any restrictions stopping a felon from obtaining a hunting license. And NYS doesn't seem to consider an airgun a firearm, so there doesn't seem to be a reason why a felon can't hunt small game with airgun and a small game license.
IN either case he's going to speak to a lawyer, but I think he should be good to go.