There is no real grocery store on WDW property, but depending on how
much cooking you're going to be doing, you may find what you need is
available right in the hotel gift shop. Many basic groceries can be found right in your hotel gift shop or food court. All of the hotels have on-site places to buy milk, soda, beer, snack foods, fresh fruit by the piece, and individual packages of cereal.
If you're looking for a bit more than that, all the resorts that have villa rooms (Contemporary, Wilderness Lodge, Animal Kingdom Lodge, Old Key West, Boardwalk, Beach Club) also have mini-convenience stores
in their gift shops. The selection is not large, but you will find more
basics like eggs, dry pasta, canned soup, a loaf
of bread, sliced cheese, etc. These resorts are all easily accessible via Disney transportation. For my family's tastes, I'd
be able to find the makings of a basic breakfast or lunch there, but
for dinner I'd need another source of supplies.
To get fresh meat or veggies, you would need to rent a car or take a
cab if you wanted to visit a grocery store. Another popular alternative
is to arrange for a grocery delivery to your hotel (you don't even need
to be there when the delivery arrives.) You can read more about this
HERE.
Have a great trip to WDW!