We just came back from a 7 day eastern where we stayed at WDW for four days before driving to Port Canaveral and then dropping off the car at Budget.
I did drive to the port first, dropped off the family and luggage, but my wife wasn't able to check us in because I made the reservations and the credit card used was under my name so the CM refused to process any paperwork until I got back from dropping off the car at Budget.
The Budget rental center had about 20 people waiting in line inside at the counter and about that many or more outside waiting for the shuttle vans. At the time they had 3 shuttle vans running back and forth to the port so once I was finished at the rental counter checking the car in I only waited about 10 minutes for a shuttle.
I do suggest if you don't need a receipt to give the rental agent your key, rental agreement contract, and odometer reading and you won't have to wait to check in your car.
Jeff