I'm not going to do a state-by-state check. However, here's a reference for Louisiana:
Department of Wildlife and Fisheries – State of Louisiana
Alligator Regulations
I. Possession
I. d. No person other than a licensed alligator farmer or licensed nongame quadruped exhibitor shall possess live alligators at any time other than by a permit issued by the department upon request for use in displays and educational purposes, and by holders of valid department issued permits for scientific purposes. Live, farm raised alligators and their accompanying alligator hide tags may be held for processing by a properly licensed alligator skinning facility without a license or permit. Violation of this Subparagraph is a Class Four violation as described in Title 56.