I have used several area grocery delivery services, and with most of them, as long as you give them a 36 hour notice, they can fulfill your order. You can place your order as far in advance as you'd like, however, you would do better to order closer to your vacation so that you can take advantage of online coupons and specials. Also, if you order too far in advance, you run the risk of an item you want no longer being available.
Some services, like
www.WeGoShop.com offer limited availability on same day service, but they recommend that you place your order a week or so before your vacation. Most of these services are a first come, first served type of scheduling, and they will usually ask for a two or three hour window during which they can deliver your items. If you want later in the day or early in the morning, you would do well to call at least a week ahead so you can get on the schedule. This helps you plan your theme park days.
Have a wonderful vacation!