We rent a car, so we are able to go to the Publix on SR192.
Without a car, it's a little tougher. You could call a cab to take you to the grocery store, if you plan on buying a lot of stuff.
All the gift shops sell groceries, but they tend to be expensive. Again though, if you are just planning on the basics, it's probably the easiest way.
Within WDW property are 2 Hess Express gas stations. One is located across from DTD. You could feasibly take the bus to DTD, walk across the street to go shopping and then take the bus back to your hotel. Again, though, you probably couldn't buy bags and bags of groceries.
The other Hess Express is located across from the Boardwalk hotel. Again, you could take a bus from your hotel to a park, change buses to the Boardwalk Hotel and then do the reverse when you were done with your shopping. The Hess Expresses have a decent selection of stuff.
There are also grocery delivery services in the area. I don't know much about those.