We did this when we stayed at GCH in March.
I used Albertson's. Their website worked well, once I figured out that I could order so specifically, like 3 bananas, 4 apples, one container of peach yogurt and 2 containers of blueberry yogurt, specific brands and sizes. It was great.
You enter a delivery address. This is the address of the hotel you are staying at. This is perfectly cool with them; they are used to doing it and the hotels seem to be used to accepting the deliveries.
You pay with a credit card, and that has your home address on it. It has to be U.S.; I've heard they won't allow Canadian billing addresses, for some reason.
You pick out what groceries you want. For example, we had milk, soymilk, juice, yogurt, pop tarts, apples, bananas, containers of "cereal to go" cups, a 24 case of water (much cheaper than the $3 a bottle you'll pay at the hotel and parks! And the tap water was horrible!), a roll of paper towels, and a small box of plastic spoons for our cereal. You can order beer and wine, too, though I've heard conflicting reports on whether you need to be there to sign for it in the case of receiving alcohol.
If you have a club card for the grocery card, you input your number and you get the same discounts you would as if shopping in the store.
There's a cost of $9.95, I believe, for delivery. You might be able to find discount codes, or it might be waived for first delivery or for a total of over $50, depending on the store (Von's or Albertson's). Personally, I didn't care; it was so nice to just have it delivered and ready with no fuss for us, it was well worth the delivery fee.
When you complete your order at Albertson's, they show you the timetable of when you want it delivered. I seem to remember it was in 1-1/2 hour or 1-1/4 hour time slots. You DO NOT need to be there; the hotel will take delivery for you. Just call the hotel ahead of time, and let them know that you have groceries coming. I have heard that some hotels even took the groceries up to the room and put them in the fridge for you, but at GCH they just held them at the bell desk for us.
I was concerned about the timing; for example, our flight was supposed to get in at 4, so I had the delivery time slot of 4-5:15, I believe. But it was delayed, and we didn't get in until 5, and therefore didn't get to the hotel until nearly 6. However, Albertsons had packed our groceries snugly in a cardboard box, and the milk, juice and other refrigerated stuff was still quite cold when we picked it up. So that was a relief. Personally, I wouldn't want to wait around in the hotel room for the grocery delivery while we were raring to get to the parks or to dinner, so I tried to time it so that it would be there by the time we got there, or very soon after.
It worked really well for us and we used just about everything up over the course of 5 days. We liked having breakfast in the room, so we didn't need to take time in a restaurant or a counter service when we could be heading to the parks.
Hope that helps!
