Hi Johanna!
There is a rental car company located at the Car Care Center on Disney World property.
The Car Care Center features an Alamo/National car rental service. A shuttle can be arranged for transportation
to/from your resort if you wish to rent a car. The shuttle does require
that you call ahead to make arrangements with at least an hour notice (407-934-4930).
In addition, there is another Alamo/National car rental located at the Swan & Dolphin resort, if this is more convenient.
The easiest way to reach the Dolphin hotel is to take bus transportation
from your resort to Hollywood Studios. After you arrive at the Studios,
do not enter the park, but instead go to the boat docks and grab a boat
ride over to the Dolphin - or walk if you are up to it (it is a
manageable walk, but long).
When looking at both of these rental agency's websites, I am able to
price out rentals that pick-up at the Car Care Center and drop off at
Orlando International Airport. It does appear that prices are much
higher for multiple destination pick-up/drop-off locations than
renting/returning to the same location.
Moms panelist Diane C give some good tips on renting from the Car Care Center. Her tips can be found
here.
Hope you have a great time at WDW and exploring Florida!