Hello!
Wow, you have two set of twins! How cool!! I'm sure they're going to have a blast.
Great question about the strollers. You can
rent strollers (singles or doubles) at any of the four Walt Disney World theme parks. Upon entering the theme parks, just follow the signs to the stroller rental areas, which will be just a short walk from the main gates.
When I rent a stroller at Walt Disney World, I opt for the length of stay option, which is the best value. I prepay for the number of days that I'll be there, and then I just show my receipt to the cast members when I enter the theme parks and I'm all set.
(FYI - the double strollers are rated for a combined weight of 100 pounds, so your 6-year-olds might be a bit too big for these depending on their size.)
Let me go on the record and say that I wouldn't hesitate a second to purchase one of these types of strollers if I could. They push so smoothly and they turn on a dime.
(If you're into cars like I am, you'll probably notice that the wheels on the Disney strollers have a negative camber - that's why they handle like race cars.)
(If you're not into cars...sorry for geeking out there for a second!)
Here's a little tip from your buddy Derek:
If you decide to rent a stroller from the theme parks, try tying a balloon or something else (e.g. a ribbon) on the handle so you can easily spot it in the stroller parking areas. Remember, your stroller may not be in the same place when you return to it after experiencing an attraction.
(The cast members constantly re-organize the stroller parking areas so they're easy to navigate as guests come and go.)
I hope that helps and I hope you have an awesome vacation!!
Best regards,
Derek