There is a mini convenience store right inside the main lobby gift shop at the Beach Club.
The selection is not large, but you will find some basics like milk,
juice, eggs, bananas, butter, yogurt, cereal, dry pasta, canned soup, a loaf of bread, sliced
cheese, 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. You can read more about this
HERE.
Have a great trip to WDW!