Igloo brand MaxCold cooler BACKPACK. Target. Go. Buy.
As for strollers being the place to keep stuff, all it takes is having stuff stolen from your stroller one time to realize that one should never have been using it as a place to keep your stuff in the first place.
Or once you've seen a squirrel rummaging around, gnawing on bottletops, snagging any food or food containers they can find, to realize that you should never have even left a water bottle in the stroller. Because ew, squirrel spit.
My son never used a stroller at home except for actual bonafide jogging excursions taken by me (with a bonafide jogging stroller), but we had one (not the jogger) for Disney. Disney is a million times different than home. And you have to think about the personalities of the adults. If you are "go go go" people (like me) and you like the stroller because even when she's tired you can keep going, you might consider keeping teh stroller. If your disney trips are all about the kidlet and you are totally 100% willing and able to take things at her pace (not something I can do easily), then try it out without the stroller. But maybe have it at the room just in case that first day goes differently than expected?
Despite him not using a stroller at home, DS needed it still for
Disneyland trips through 5. Then we had our first WDW trip when he was 6.5, brought the Volo, and left it up to him. It never even got unfolded; just stayed in the hallway of the villa the whole trip. He never wanted it.
Thankfully we had had the experience of theft from our stroller twice (the second time was absolutely nothing of value, but it still went to show us that thieves will take anything) and I'd seen the squirrels already, so we never took anything we couldn't carry and never left even a water bottle (thank you, MaxCold cooler backpack) in the stroller starting when he was 4, so there was no difference to use when the stroller was gone.