Hi Ikeyla! When traveling with children it is best to prepare and plan ahead of time so there will not be any surprises to deter from a great vacation.
First let's talk about the double stroller and buses at Walt Disney World. Most of the complimentary transportation will be by bus and it is state law that requires all strollers to be folded down as small as possible before loading on a bus. Double strollers are extremely bulky and difficult to do this while trying to handle all the bags and children. But there is not an option to this as it is a law. I have seen many guests arguing with the drivers who are simply trying to do their job.
Some boats will handle the larger strollers without breaking them down while the smaller boats will not allow open strollers or wheelchairs and ECVs. It is simply a safety issue and again not negotiable. Monorails will allow strollers to remain open but the double strollers take up a generous amount of room in a car. I have never seen a taxi or a taxi van not fold down a stroller so my guess is that all would require that.
My grandson has walked the parks since he was four and although there were times he was tired we made sure we went at a pace he could handle and had plenty of rest times. If you don't think your oldest could handle the walking then consider
Renting a Stroller for your time in the parks where you will do most of the walking. These must remain in the parks so you won't have it for the resorts but if you need one there then consider renting from an outside stroller company. They will have many options of types such as the step on behind for the older child which ultimately would be easier then a full double stroller. Kingdom Strollers and Orlando Stroller Rentals are two companies that can deliver and pick up from the resorts making it an easy transaction.