Hi Heather,
I'm so pleased Tracey recommended the panel I hope I can be of some help!
You're right, quite a lot of American guests do mail a parcel to be waiting for them at their hotel when they check in. Personally I don't think this would be a good idea from England, I recently sent a small bar of chocolate (Flake) to an American friend and nearly fainted when it cost almost £2 to send so a big parcel would cost quite a lot. Don't worry though as Disney does stock quite a bit of basic baby care, including nappies (diapers) but what I actually recommend you do is have a look at a website called
www.gardengrocer.com They offer a great service where they shop for you and deliver it to your hotel. I just had a look and they offer 73 different types of diapers so you should find what you're looking for. They also sell perishables such as bread and milk too which could be handy and some yummy American candy!
Another option is to make a trip to a supermarket such as a Walmart, you can get a taxi to one if you don't have a car. To be honest though the delivery charge with garden Grocer will be quite a lot less than the taxi fare and you won't need to use up any park time to get your shopping!
Also be sure to check your airlines luggage allowance, I know we never use up all of ours, (I know that VirginAtlantic allows 2 pieces per person) so you could maybe pack a holdall with non-breakables such as the nappies in!
Have a fantastic holiday,
Jo