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.
All the resorts that have villa rooms also have mini-convenience stores in their gift shops. The selection is not large, but you will find some basics like milk, juice, eggs, cereal, dry pasta, canned soup, a loaf of bread, sliced cheese, beer, soda, etc. 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!