Hi there, Hallie!
While the wagons, as you note, are not permitted into any of the Parks at
Walt Disney World, there are no prohibitions against having them at the Resorts.
I'd offer two suggestions:
1 - Stroller rental. While you can
rent strollers at Walt Disney World, you can look into some of the local companies that offer that kind of service. I've heard great things about www.OrlandoStrollerRental.com from lots of satisfied customers (including some of my fellow Moms Panelists). These can be taken back and forth to Parks, around your Resort, etc.
2 - Purchasing an inexpensive umbrella stroller from a shop on property or from a local big box store. That way, you aren't committing to anything particularly elegant, and if it gets ruined - or you choose to leave it behind - there isn't much in the way of investment lost.
Have a wonderful time on your trip!
See you later, celebrator,
-Deborah B.