Well, first, I have had things stolen from my stroller, so I don't bring anything that I can't and won't carry. That happened at
Disneyland, but WDW is bigger with more people, so I imagine the possible criminal element is higher at WDW. Don't bring anything you can't carry, and take into rides, with you! That way you don't have to deal with the violation of something being taken from you.
And it also releases YOU from the need for the wheels.
We had a jogging stroller *for jogging*, and we had a Volo (and before that, some other stroller that he outgrew almost immediately) for Disney. DS walked, was carried, or was worn for EVERYTHING else from the very beginning (OK obviously he didn't walk from the beginning). I took the Volo for our first WDW trip when DS was 7, because no one knew what it was going to be like for any of us! He had needed it the year before at Disneyland. We went out the first day without it, to see what it was like. And he never once asked for it.
Sure, there was some soreness, and he and I (and my mom before us) suffer from some weird genetically-passed-along deep bone leg-ache problem, but we dealt with those problems with arnica (and epsom salts) and trying to slow down a little.
He really amazed me, that's for sure! Maybe your child will amaze, you, too! Of course, it's easier if you still have something small and easy to take alone like a Volo.