Strollers can be rented upon entering the theme parks and at Disney Springs. Strollers are not available at the water parks.
The Disney strollers are made of hard plastic. They do not recline but they do have canopies for some protection from the sun. (We do not recommend the Disney strollers for infants.) It's a good idea to bring either a receiving blanket or beach towel to provide a bit of cushioning between the plastic and your child.
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If you regret not bringing a stroller of your own, you'll find that they also have a nice umbrella stroller available for purchase. It's priced around $40 and can be found in the vicinity of the rental areas.
Walt Disney World Stroller Rental Fees
When purchasing a Length of Stay rental, you show your receipt and can bypass the queue with little or no wait.