We used the Disney grocery service (faxed our order ahead of time) and they delivered the items to our room on the day of arrival (after 5pm, if I remember correctly). Had no problems with it. Of course, the items are pretty expensive but are the same price as what you would purchase in the grocery section of the hotel store (plus a delivery charge of $5.00 I think). I would use the service again but it is not a "must have" for us. Just a nice convenience.
We never used the Garden Grocer service, so can't help you there. But I do remember seeing references to the service on these boards somewhere.