SS Norway (12 times) '85 - '97; Radiance OS 06/01; Island Princess 11/04, Grand Princess 11/04/05, Golden Princess 7/21/06, Golden Princess 2/24/07, Explorer of the Seas 12/23/07, Island Princess 11/15/08
Yes they do, and any of the big name hotels around downtown has a policy regarding this. Most are going to charge you something to leave your car on their property. But, the hotels around downtown tampa realize and focus their marketing around the port.
If you have extra time to kill or looking for a fun place close by, Channelside is a nice spot to take the family. It is located right next to the port entrance.