Hi, Christi!
You will see strollers all over Walt Disney World. LOTS and LOTS of strollers! Because the Cast Members are used to the volume of them, you will actually see "Stroller Parking Areas" for you to park your stroller while on an attraction or in a restaurant.
While there is no need to lock it up (in fact, it is preferred that you don't as sometimes one stroller has to be moved to reach another) I do suggest you keep all valuables with you - never store your purse, phone, or camera in it while you are away. The parking areas are NOT covered, so you may want to bring a rain poncho or small tarp to cover it. An afternoon rain in Florida is not uncommon, so cover your buggy anytime you will be leaving it.
I suggest you bring something to identify your stroller. They all start to look alike so a balloon, ribbon, or sign can make a big difference. My favorite was I saw a mom once who got battery powered Christmas lights and had them wrapped around the handle (out of baby's reach) which made her stroller easy to find and festive all at the same time!
You are welcome to bring your own
stroller from home, or you can rent them at any of the theme parks.
Enjoy your magical
Walt Disney World vacation!
Cathy