The strollers can get expensive to rent, but IMHO, their biggest drawback is the fact that you can't take them out of the parks. Staying at the CR, it's so easy to be able to wheel your (sleeping) kids onto the monorail, off the monorail and all the way back to your rooms.
For AK and MGM, yes, you will have to fold up the stroller and get it on the bus with you, but the walk from the front of the park to the bus (where there are very few seats while you wait) and then from the bus to your room (at most resorts) can be verrry looong. (Not to mention getting around the resorts while you're there.) So I'd recommend BYO stroller, but make it the umbrella stroller, so it's easier to take on the busses and attractions with you.
I'd also advise against the double stroller, as long as you have 2 people who can push strollers. Doubles are bigger, heavier, more difficult to get on and off transportation and more diffcult to navigate through heavy crowds.
Have a great time!