I've read here that strollers are periodically vandalized or stollen, but in our MANY trips to WDW, with extended family and often multiple strollers, it has never happened to us. (nor at any other amusement parks)
Even with a stroller, we tend to pack light. We have found that the best course of action.
Mostly we only leave non valuables in the stroller- bottled water, one day supply of diapers/wipes, maybe a change of baby pants, jackets, stroller rain cover or plastic poncho as makeshift rain cover.
If you can, on your person, carry all forms of money, ONE diaper, tiny quantity of wipes, tiny hand sanitizer, ID, camera, cell phone, emergency meds, and other essentials. (for the guys, cargo shorts are great- they can carry about half these items for you! esp water)
There are times when you will have to park the stroller outside a pavillion - think Epcot- but you'll be in that pavillion fo a while- so being able to change a diaper without having to go all the way back to the stroller parking area is handy. Conversely, we don't want to lug around ANYTHING extra. Extra stuff reduces your endurance.
Once the kids are out of diapers, we pretty much only put water and maybe sweaters/jackets/rain jackets in the stroller. Oh - and non-valuable stuff we've bought. Oh - and once in a while- flip flops for riding Kali RR.
One thing that you may find handy is a way to make your stroller STAND OUT fomr teh rest- stickers are bright, inexpensive, and often tricky to remove. You can also tie (knot!) something BRIGHT around the handle to make it stand out - and hard to steal. You willl want to do that if your stroller is a common stroller color (blue, grey) and/or very expensive, and/or brand new, and /or a very common brand, etc. If there's any temptation to steal - I figure it's going to be the most expensive newer strollers, in neutral colors. At least that's my way of thinking.
That and a small child could be tempted to walk off with an obvious stuffed animal. If you have any toys that are irreplacable from a sentimental pov, I wouldn't bring those with you either. Almost any gift shop will give a shopping bag, we hide toys in a bag to reduce that sort of temptation.