Gotta remember that the beaches are actually part of the State road system. Galveston County will issue a license plate to a golf cart if the owner puts the slow moving vehicle triangle and the appropriate lights on one. Don't know if that applies to all counties, though.
DPS might be able to fill you in on exactly what it is you will need.
Also, remember that the State speed limit on the beaches is 15 mph. Can't go zipping along at 40+ mph like you can on private property (even though it's pretty darn good fun!). Could get you a pretty hefty ticket should a law enforcement type actually be there and decide to pursue the matter.
Good luck finding out and have fun!