There are busses to the Disney grocery stores, if you want to call it that. The normal grocery stores are off Disney property. The two closest are Gooding's and Walmart. Gooding prices are outrageous.
Since you won't have a car, can I recommend wegoshop. They delivery to your disney resort www.wegoshop.com
We used the service in the past with the previous owner and now are set up towith the current owner for a delivery this next trip. The owner called and arranged the set up with us after filling out the online request. She is very nice.
We have used Garden grocer (also another local delivery service www.gardengrocer.com) and they provided good service as well.
Not sure about how often they run but one of my books says it is 25 minutes to MK 10 minutes to Epcot 10 minutes to MGM 15 minutes to AK and 5 minutes to Downtown Disney 15 if you want to take the boat.
Hope this helps!
WELCOME!