Unfortunately, wagons and stroller wagons are not permitted in any of the Disney parks as of 2019. This was largely in part to issues with the wagons causing congestion in the park walkways.
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Are stroller-wagons permitted? Stroller wagons, whether pushed or pulled, will not be permitted at any theme park or water park or at indoor venues at ESPN Wide World of Sports Complex.
To make sure you are fully aware of Disney's stroller rules and guidelines, and to take out all the guesswork, here is what you need to know: Strollers heading to a Disney park must be no larger than 31 inches (79 cm) wide and 52 inches (132 cm) long. Stroller wagons are not allowed.
This was largely in part to issues with the wagons causing congestion in the park walkways. The only exception to this rule is if the wagon is deemed medically necessary for the child(ren).
Wagons, including stroller wagons, are not permitted. If you need a stroller, rentals are available just outside Disneyland Park to the east. Double strollers are also available if you need space for two children or for holding more gear.
Strollers over 20 lbs must be checked at the ticket counter. Wagons, cribs and pack 'n plays count as regular baggage and must not exceed the carry-on weight requirements. If they do, they will have to be checked and incur regular fees. Only one carry-on diaper bag per child is allowed.
Most zoos allow Wonderfold wagons and stroller wagons in their parks even on their high-traffic days. Zoos typically offer wider pathways thanks to being outdoors and, depending on where you live, your zoo might not have many “high-traffic” days to begin with.
Unfortunately, in alignment with the Disneyland Resort Rules, “Wagons, including stroller wagons” are on the list of prohibited items. If you don't have a different stroller at home that meets the park's guidelines, then I recommend renting a stroller there.
Collapsible wagons may be brought onboard. However, they are the only type of wagon permitted, and they must be stored in your stateroom or the designated parking area on Deck 2 (midship) for the Disney Dream or Deck 6 (midship) for the Disney Magic when not in use.