When we were there last spring, we just found it much easier to leave a few extra diapers, a change of toddler clothes, ponchos, sunscreen, etc, in the bottom of the stroller, knowing they were potentially at risk. We were lucky and nobody seemed to bother any of it. One thing to consider though, is that you have to get those things into the park so after the first day, I used one of those clear Disney bags from a store for going through the bag check and then "set up" the stroller for the rest of the day. That way the bag we carried on rides, with wallets, cameras, etc. wasn't jam packed when we went through bag check and they usually just kind of scanned what was in the clear bag (although I understand they may have stepped up security since then) and then could check our backpack much more easily.
The other thing that I consistently, MISTAKENLY, left in the stroller was my phone. With the ability to check ride times and change fastpasses on the fly, I was constantly on my phone (a frustration for a different post but anyway....) and because I usually wear sundresses without pockets I would make changes while we were walking and absentmindedly drop the phone into the parent console on the stroller with the intention of grabbing it when we got to our next stop and transferring it into the backpack my husband was wearing. I can't even tell you and would be embarrassed to know how many times I left the stupid thing in that little square pocket of the stroller console and ended up rushing out of restaurants and attractions to retrieve the phone. I was so lucky that nobody ever grabbed it. That would really have been a lesson learned but I'm lucky that I didn't. That little square part of the console has like a flap that covers it so I suppose it was out of sight and out of mind as we rushed to the next fun thing but it still would have been so easy for someone to stroll by and grab the thing. Ugh, my blood pressure rises just thinking about how many times we panicked over that.