My family has stayed at the Beach Club Villas many times. It's got the perfect combination of amenities and location. A real winner on all counts.
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!