When traveling with young children it is always a struggle deciding what to do about strollers!
Bringing your own means checking it with your luggage at the airport. Large double strollers will not fit in overhead areas. You can have it checked at the gate but be diligent in making sure it gets onto the plane.
Each Disney park has double stroller rentals available for guests. They are hard sided but efficient. Many moms don't like that they do not have soft interiors for their children to nap comfortably. If a simple blanket on the bottom would not be sufficient then it is probably best you bring your own that you are familiar with. Double strollers rent for $31 a day but there are small deals when you rent for the length of stay. These strollers cannot be taken back to resorts.
If you do decide to bring your own stroller please be aware that when riding on the Disney transportation busses the strollers must all be folded up completely. This is a Florida law and drivers will be quite firm that all are secured before they can leave from that stop. I have seen many families really struggle doing this when they have used the stroller to carry and hold every item they have. Most strollers will not fold with anything stored in them thus everything must be emptied out.
There are a couple other options you might consider. You can also rent strollers from local companies. One that has had good reviews is
www.orlandostrollerrentals.com . They have a variety of choices and will deliver to your resort. This is an excellent way to get the kind you prefer without the hassle of flying with your own.
Disney also has small single umbrella strollers available for sale at most resort shops and within the parks. They are made by Kolcraft and retail for $53.20 including tax.