We just used budget two weeks ago. When you get off the ship, tell the porter you are using Budget to pick up a car, and they will direct you out to a white canopy area in the parking lot. The budget van will come by that location to pick you up and take you to their rental office.
We found it worked best if only one person does this and the rest in your party stay at the terminal with your luggage. When the person picks up your car they can then come and get you and your luggage.
When my husband went to get the rental car there were two entire families loading all their luggage in the van used to transport people to the rental office. It filled the van up and made everyone waiting take longer than needed.