Sara,
Many, if not all, of the Disney resorts have publicly accessible washing facilities available. They even vend soaps, if needed.
If you write me back with a follow up question telling me which resort will be your vacation home, I can get back to you with more information of where the facilities are located, and confirm that your resort does have them.
Also, while they do vend soap there, you might consider ordering from a grocery delivery company in the Orlando area so that you have exactly the kind of detergent that you like to use. If you don't have a car, these services can be a real live (and budget) saver!
One nearby grocery store, Goodings (
www.goodings.com) has a minimum order requirement, as well as a fairly significant delivery charge. but if you are ordering a lot, it might be worth it.
Another choice is
www.gardengrocer.com. It appears there that you have a lower delivery charge, and can get that waved with very large orders.
This way you can get those baby supplies, including detergent, for your room.
Have a magical vacation!