Try these 3 local companies in Galveston, they all provide transfers for different budgets.
Galveston Shuttle & Limo: http://galveston-shuttle.com/get-to-know-us/rates/ (private car transfers)
Galveston Express: https://galvestonexpress.com/galveston-taxi-rates-schedule/ (Shared shuttle Service)
Island Breeze Shuttle: https://www.islandbreezeshuttle.com/rates/ (private car transfers)