I'm sure the limo driver will stop at Publix if you ask, it's only a few minutes from the RPR. Of coarse, I'd tip the driver well if they do it.
It's a very good idea to stock up on groceries before checking into your hotel. The RPR doesn't have refridgerators, unless you ask, so buy a styrophome cooler in the beer isle, and put your drinks in it. Take it to Disney with you, and restock it. Then when you check out, leave the cooler next to an ice machine.
Yes, carrying your groceries to your room may take two trips, and the supplies will be heavy, but once you get them into the room, you'll be happy.