Hiya Elizabeth,
Thanks for stopping by planDisney. It wasn't long ago that I was in your shoes - taking
three babes under 3 (and all the accompanying gear) to the parks. My daughter and twins are 21 months apart, so we spent a lot of time in strollers!
So, while I empathize with you,
the rules for the 'World are very clear about wagons (including stroller wagons) - they are not permitted. You can bring any stroller smaller than 31" in width and 52" in length. You also have a couple of options for
stroller rentals. You can rent single or double strollers in the parks for a day or the length of your stay. You can also look into renting a stroller through an outside company. ScooterBug Orlando is the preferred resort provider. We rented a double stroller from them during one of our visits, and the process was straightforward.
We would usually bring the boys' double stroller and then rent a single stroller for our daughter or bring a small umbrella stroller (she was much easier to contain!). I took all
the kiddos to Hollywood Studios on my own once (!), and I wore one of the boys in a carrier and put his twin and my daughter in the double stroller.
I wish you the best of luck! If you have any other questions, please don't hesitate to come back.
Rayna