Take a Disney bus to DTD, get off at the PI stop and walk over to the Hess mini-mart. Good pricing on soda, water, beer, "fun food" and some "real food".
I guess you could walk from DTD to the Goodings (Crossroads) but Goodings is very expensive and you probably would want a taxi to go back. How many bags of groceries do you want to carry?
I'd go with the Hess mini-mart, use a grocery delivery service or rent a car.
edited to add--If you don't have to worry about checking paying to check luggage you could bring stuff like coffee, ceral, bagels etc. Purchase pershiables like milk in your resort. Unless you have to buy a lot of items paying resort pricing may make more sense then paying for a delivery service, taxi or car rental.