The cheapest route will be for you all to use the FREE DME and then get a grocery delivery from wegoshop, gardengrocer, or Gooding's. I have no experience with the first two, but Gooding's charges a flat $10 delivery fee with a $50 minimum order. A delivery will also save your husband a lot of time -- getting a rental car then driving to a store in an unfamiliar area then shopping in an unfamiliar store then driving to the resort and unloading then driving the car back to the airport then catching a DME bus back to the resort is A LOT MORE TIME CONSUMING than taking the DME bus with the rest of the family and getting a grocery delivery.
Again, I don't know about wegoshop or gardengrocer, but with Gooding's you don't have to wait around for your delivery or schedule a time for it. They deliver to your resort's Bell Services (BS), and then Bell Services calls you. If you're in your room, BS will bring your groceries soon. If you're not in, they leave you a message and then they store your groceries for you and even keep the cold stuff cold (Gooding's separately bags and labels the cold stuff).