Hi, Cary! Thanks for visiting us again here on the Disney Parks Moms Panel.
Happy first birthday to your youngest princess! What fun to be celebrating at Walt Disney World!
Yes, you can rent a double stroller at each of the four theme parks. The
Disney rental strollers are quite a bit larger than a standard folding stroller and can definitely accommodate a 4-year-old. The rental strollers are constructed of hard molded plastic (there is no padding) and they have a nice canopy overhang for shade. There is a small pocket in the back for storage and two cup holders.
I absolutely agree that having a stroller is essential to helping prevent meltdowns! Strollers provide a quiet place to rest and save little legs from tiring out. Since there is a lot of walking required when visiting the theme parks, many families use strollers for their children while on vacation even though they may not use a stroller any longer at home. We used a stroller for my daughter at Disney until she was nearly 6 years old (though she had stopped using a stroller at home before age 4).
You may want to keep in mind that when you're leaving a theme park, you'll have to return your rental stroller near the entrance. This means that you won't have a stroller when you travel from the theme park entrance back to your resort hotel room. Due to this situation, I have personally always brought my own Maclaren folding stroller from home and simply gate-checked it before boarding the plane.
Wishing you a magical vacation, Cary! Please write back with any other questions as you plan your trip.