Hi, Claudia! To maybe point you in the right direction from the start, the strollers available to rent at Walt Disney World don't leave the theme parks, so you won't have to worry about them on the buses or boats back to the resorts or in between parks and restaurants. If you think you might need a stroller to get the kids back to your resort at night, you might want to consider the outside agencies that rent them.
There is a lot to be said, though, for the theme park strollers, especially with older children. The double stroller is a good size, and the hard plastic makes it a great "seat" for things like parades and outside shows. You won't be able to recline it, but at least the little one will be able to sprawl out on it and get a quick nap in.
You'll find a place to store some of your essentials in the Disney stroller, although I will say that some of the rental strollers may have a larger basket or more cup holders.
The cost to rent a stroller at Walt Disney World can be found by clicking
Walt Disney World Stroller Rental. You'll see prices starting at about $13 per day for a single stroller, or starting at $27 for a double.
I hope this helps!