The first time we went, DD & older DS were 5 & 3, & we brought 2 Chicco umbrella strollers, & we used them the majority of the time probably. But there was also a lot of times that we parked the stroller (like in Fantasyland) & did all the rides before we went back & got the strollers.
The next year, they were 6 & 4, & we ended up leaving them at the resort after the 1st day - they just didn't need them. I think we might have also taken them w/ us on our EPCOT day since it's a big park w/ a lot of walking.
Younger DS has been to WDW when he was 11 months, 2 years, & 3 years. So far, we've used an umbrella stroller for all 3 trips. (One of the same Chicco strollers from our bigger kids' stroller days - the Chicco strollers are really durable!

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However, last year, he didn't use the stroller all the time, & we parked it several times while he walked (like in the different lands in the MK, between The Land & Living w/ the Seas at EPCOT, etc.). And there were many times, he preferred walking to riding, & DH or I pushed an empty stroller.
This year, we'll take it w/ us (he'll be 4), & I foresee us using it even less.
I do like having the stroller as a place of "storage" for all the extra stuff - water bottles, autograph books, maps, clear bag w/ a couple of diapers & wipes, etc. I also loop my bag over the handles rather than carry it when we're pushing the stroller as well.
So I'll miss that aspect of a stroller... like the PP, though, I'll just carry less stuff!
What I will NOT miss is pushing a stroller through the crowds!! You can get through the parks so much easier & faster w/o a stroller - especially after the fireworks shows! And weaving around & dodging people who jump in front of your stroller is the pits!
Last year, for Fantasmic, we actually took the stroller back to the van, so we wouldn't have to hunt for our stroller in the masses of people after Fantasmic. We were able to get out of the crowds so much quicker w/o the dang stroller!
In busy crowds or masses of people (like after the fireworks shows), DH just puts younger DS on his shoulders.
So, yes, younger children probably need strollers, & it's nice to have a stroller for the extra storage. But I would say most children over 5 can handle the walking just fine. You might miss the storage you gain w/ a stroller, but, trust me - you will NOT miss finagling a stroller through the crowds!
Oh, & on the buses too - strollers are a pain!
Another note - While WDW might be a lot of walking, it's not really a of sustained walking. You walk from place to place &, then once you get to wherever you're going, you either stand or sit.
So, yes, at the end of the day, your feet are sore, but it's not really marathon walking or anything.