There are 2 ways to do grocery deliver. You can arrange to have a DVC grocery service, which is basically items available in AP. I think the members website has a price list, as do some other DVC fan sites. Supplies are limited and pricey becuae they come just from the grocery area of the gift shop.
We always use gardengrocer.com, which a lot of people enjoy and get great service from. I think delivery fee is $10. Some find it pricey, with minimum order and fee, but I find it cheaper than other options and not out of line with what I pay locally. If you pay online (Paypal or credit card, I think), they'll deliver to bell servcies so you don't have to be present. Bell services can keep items cool and will deliver to your room upon request. We've done this twice and have had no problems. The owner even calls our cell phone to leave a delivery message that they're there.
Others also like wegoshop.com. It's a more personal service, and people rave about it, but it's too much human interaction for me. I just want to order my items and have them delivered without worrying about being in a particular place at a particular time!
I vote for gardengrocer. We used the resort one time. It was a little expensive and we could not get certain items. They get it right out the store at the resort.
We have used gardengrocer many times and love it. We have never had a problem with them. Bell service delivered the food to our room for us. Hope this helps.