At all the Disney Resorts they have little stores in them called sundrie stores. They all have milk, juice, water, fresh fruits, breads, cereals, snacks, chips that sort of thing. They also have suntan lotion, diapers and such. Sure the prices are a little higher than Walmart but if you don't have a car to get there and rent one, then you really aren't saving much. Another option is ordering some groceries and having them delivered to the resort(don't need to be present when they are delivered....FD will take them and refrig/freeze anything that need to be kept cold). Here are two stores that I know deliver to the resorts.... goodings.com and gardengrocer.com
If you have reserved a family suite at DAAR (Nemo,Cars or Lion King) then you will have a mini-kitchen in the room. It has a small fridge (with small freezer), micro and paper plates and plastic silverware will be provided. If you have a Little Mermaid room then you won't have a mini kitchen but the same fridge.
Since you have the dining plan already (was going to suggest that) don't forget that you get a free refillable mugs with that. You can only refill it at your home resort and they only offer coffee, pop and hot/cold tea and hot cocoa...no milk
Have fun at Disney World, it truly is a magical place....
N.