You'd have to use a LOT of shampoo, conditioner, etc. to run out by day 4. They are small bottles, but can be made to last. If you feel that's not enough though, you can buy such items in the little sundries shops in the resorts.
I don't think it's a bad idea, assuming you have room, to go ahead and pack some paper plates, plastic dinnerware, etc. Those don't weigh that much and should present no problems with baggage handling. I'd stay away from trying to bring any liquids, if possible, especially since you can buy those types of things there. If your baby is still in diapers, I'd order those from GG or get them off-property. They're too bulky to pack, and way too expensive on-property. We found that out the painful way one trip.
Don't forget sunscreen. If you can buy that off-property, you'll save some money. Same goes for baby formula, baby food, etc. You can find all this on-property, but off-property is SO much cheaper.
I see you mentioned you wouldn't be renting a car, but I'll say this: there is a super-easy car rental location at the Dolphin resort (easy walk from BWV). You could rent a car for a one day rental and go shopping at a nearby Publix or
Walmart for a lot less than on-property, and probably with a lot more flexibility for whatever circumstances arise that might make ordering through GG problematic. Once the shopping is done and you've offloaded the groceries to the villa, drive the rental back over to the Dolphin, hand the keys to one of the valet people (tip him a few bucks), and walk back to BWV in 10 minutes. I wasn't sure if you knew you could rent from there, if you wanted. A lot of people rent a car for a few hours just to do that quick shopping trip and bring it right back. We've done it a couple times when we used to fly down.