It is certainly difficult to travel light with children in tow!
When our son was of stroller-age, we were in the driving-to-Disney phase. Partly to save money and also in order to load up our van with all of the toddler gear required to keep temper tantrums (his and ours) at bay. We were able to bring not only our stroller but also kid-friendly snacks, drinks and familiar stuffed animals to make day and night time routines run smoothly.
While taking a stroller on a plane may be a hassle, I would absolutely recommend having your own stroller to tour WDW. Some of the most tiring times of your day will be in the early morning hours when heading from your hotel to the park and when departing a park in the evening. I could not imagine traipsing back to tour hotel at the end of a long day without a stroller. Of course, this was much easier to accomplish when we had our own vehicle and were at the umbrella stroller age.
Airline travel can also be exhausting. If your flight is delayed, the stroller is a great place to let a child get some rest. Gate check the stroller!
Stroller rentals are available at WDW, however I personally find the fees too high.
Take your own stroller. Waiting for buses at WDW is another time when parents and kids are tired and losing patience. If a child can rest in a stroller, it makes your transportation time a little easier!