Hi Jessica!
When traveling with young children it is really helpful to either ship baby supplies ahead or have them ready and waiting once you arrive. There are several ways to have this happen at Walt Disney World.
Most guests will use a grocery delivery service and my favorite is gardengrocer.com. This is a company who was one of the first in grocery delivery to Walt Disney World resorts and has a very good reputation. They carry much more then groceries and you will find all kinds of baby supplies that you might need. They do have a $50 minimum but that never seems to be hard to make. If you wish to use this service please order and reserve your delivery date at least a month ahead especially during holiday times. They can only deliver to so many guests per day and you don't want to have your date booked.
Resorts will accept deliveries on your check in date but often will not put things in your room until you actually arrive due to too many room changes that can happen when you check in. If you arrive you simply ask the Bell Services to deliver your food. The perishables will be stored in coolers so no worries about that. If you are in a regular room don't order too much that needs to be refrigerated as they are quite small. If you are in a DVC unit they will have full size refrigerators and full kitchens.
If you are a DVC member you can use the special
DVC Grocery Delivery Service. Each property will have an order form for you to choose from to put in before your arrival or after you arrive. Fort Wilderness will also accept grocery orders before you arrive but they will only offer those items that are in their Trading Posts.
Hope this helps with your planning.