Dollar has a rental counter at the Courtyard by Marriott near Downtown Disney. They don't staff the counter around the clock, so you would have to pick up your car when they are staffed (although you can drop off your car anytime and leave the keys and contract in the drop box. I think their hours are 8:30a-4:00pm. The Courtyard is not a WDW resort, but they will pick you up and drop you off at your resort. Or, since they are located only about 3-4 blocks from Downtown Disney, you can walk to/from DD and use Disney transportation to get to/from your resort.
I rented a Dollar car at the airport, then dropped the car off at the Courtyard, walked to DD and spent some time there, then took the boat back to Port Orleans Riverside. I also rented a car to go back to the airport, for that rental, we were at Blizzard Beach, so we took the Disney Bus to DD instead of the resort, then walked to the Courtyard, picked up the car and drove to the resort to load up the luggage.