Hi Carolyn!
Traveling with the younger members of the family can be much easier then you think when you are taking them to Walt Disney World. I find that the biggest issue is parents over planning what they think they will need for a day in the parks.
My first suggestion is to keep the stroller issue simple. Consider bringing only an adequate umbrella stroller versus renting something big. Disney even offers in all of their resort shops a simple yet very adequate umbrella stroller for between $50-$60. This stroller will recline, has a covered top, and even has a cup holder. It is a perfect solution when needing something to transport the youngest member of the family. It is also much cheaper then trying to
Rent a stroller from an outside company or daily in the parks.
The umbrella strollers are easy to close up for bus transportation where that is required. They are easy to maneuver among crowds, elevators, and shops. I have found with my grandchildren that they might still need a stroller but when they see something they want out and out right away. Children at that age will be in and out of a stroller to see characters or parades. When they are tired they will fall asleep anywhere in any type of stroller so catching a nap is not an issue. Keeping it simple in my family has proven to be the best solution to keeping everyone happy for a day in the parks!
Trying to maneuver two strollers will be difficult and although parks have some lockers most will not fit a stroller inside. Beyond that there are no secure places to keep things safe when you are not using them.