Albertson's delivered to us just fine at GHC three weeks ago. You give them your billing address, and then there is a separate delivery address which would be the hotel. We paid with a credit card. I don't believe they charge you until the day they deliver, because you can change your order up until 10 p.m. the night before.
They have 1-1/2 hour window delivery times. The hotel will hold your order for you at the bell desk until you get there. We expected to arrive about 5 p.m., so my delivery window was 3:30-5 p.m. (I didn't want to wait around for our groceries because I knew we'd want to get to the parks!) I worried that it wouldn't be good for milk and yogurt to sit, but it was fine, and after all it was probably only a half hour or so before it got into our room's fridge.
Hope that helps!