I haven't done it for just one day, but it's certainly possible to.
The Car Care Center on property has Alamo and National, and they will pick you up and bring you back to your resort. We used Alamo and that worked.
Budget, Enterprise, Hertz and I'm sure some others have offices in different hotels at Downtown Disney, and they will also pick you up and bring you back to your resorts. If I were you, I'd use whichever one was the cheapest for the day I needed it.
We do this every trip! We go to Wal-Mart & the outlet malls on our arrival day. The next morning we drive to the resorts to look around & eat a TS breakfast (usually at Boma ) I have allways rented from the car care center except once.
We do it every visit too!!! Same thing go to Walmart, the outlets and then take it back to one of the hotels. So this year we are getting it from Alamo, Car Care Center...it is $33.60 for the day. I used code 299851.
I've never rented a car from the Car Care Center before and decided to do it this year to do some shopping and side trips. I decided to enroll in the Emerald Club at National fo rfree (through the AllEars website) and set it up. I got a great deal and basically am spending $78 for 4 days! i'm looking forward to it. We may also take in a shuttle launch!