Just another thought on the car rental/
Walmart idea since it something we explored...
There are car rental companies at some of the hotels in Downtown Disney. We considered renting at the airport and dropping off at Downtown Disney the same day or next day.
The cost of a 1 day rental is cheap and can be balanced out easily by the savings in shipping and also being able to buy some groceries and save on breakfast cost. Also, if you had any interest in going to Sea World you could get there. If you haven't been to Sea World it's great for an infant/toddler.
In the end we decided to get a rental car because the buses (esp. to the All-Star resorts) are very slow and you'll also find that the bus stops for the all star resorts are always the farthest away...not ideal for travel with a little one and a stroller. Also, I'm not yet comfortable with the idea of him not being in a car seat. When I lived in Orlando my roommate worked for Disney Transportation and he has told me that there are many minor injuries reported from the bus transportation (usually someone falling).
Anyhow, depending on when you go it might be worth exploring the option.
I also remember seeing a few posts around here regarding grocery delivery services. I'm sure they would deliver anything that you can get at a grocery store including diapers and the like.
Either way I hope you find a situation that works well and your family will surely have a wonderful vacation.