Only you know your children best. I would probably try it without a stroller, but reserve the option of renting one at any time, if they seemed overtired or just worn out! We just returned from our trip and I almost brought a stroller along for our just-turned 6 year old. But I decided it wasn't worth the hassle in the airport and that we would rent at the parks when necessary. I also didn't know what we would do with our backpack while in the rides, as we usually leave it in the stroller!
To make a long story short, we didn't need to rent one all week! The only day it was a possibility was at Epcot, around 5pm. But by the time we thought of it we were pretty far away from a rental spot. DH carried DS on his shoulders for a bit (10 minutes) and that helped. He managed to go along under his own power for the rest of the week! Granted, there were a few late afternoons that he was dragging a bit. But a short break and snack stop usually helped that.
I will say, in the past it has been a relief to my mind, when we are leaving the parks at night, in a crowd, to know that my kids are in the stroller and I didn't have to worry about being separated from them. But DH took one child and I took one child and we agreed to meet at the car if we got separated from each other (the pairs of us). We held on tight and managed just fine!
And it was nice not to have to maneuver a stroller through some of those crowds.
The overall positive effect was that we felt MUCH FREE-ER to come and go throughout the park without the hassle of stopping to park/pick up the stroller. Also, just nice to know that DS and DD could keep up with the miles a day that we logged.
The biggest drawback, IMHO, was what to do with the "stuff" since we didn't have a stroller to dump it in. We carried a backpack on the days when the weather was iffy, in order to have our ponchos and umbrellas right with us. We kept it as small and light as possible and when we were on rides, we put it between our legs or put a leg through a strap, so it wouldn't go flying out (think RNR!).
On the days with no threat of rain, we rented a locker in the morning. I thought it would be a real hassle to trudge back and forth to the locker, but we really only needed to do it once or twice a day, to drop off or retrieve sweatshirts. Our day at Epcot would have been much more pleasant (and longer) if we had thought to put long pants and warmer coats in the locker! And our day at MGM was perfect, because first thing in the morning we put a large duffle bag full of long pants, warm jackets and beach towels (to sit on) for our evening at Fantasmic. It definitely saved that evening. And people looked envious of our comfort!
ANYHOW...... I have gone off the track here...... Only you know what your children need. Why make an expensive trip miserable just because "someone" thinks they might be to old for a stroller. Do what makes them comfortable and you will ALL enjoy your trip a great deal more!! (You wouldn't deny them water if they were thirsty, so why deny them rest and comfort when they are tired?) It is certainly a lot more pleasant than getting mad at them for not keeping up with you, or feeling resentment because you had to leave early because they were tired of walking........P