I have three kids and have used strollers on six annual trips. Reasons to get a stroller:
1) You can move much faster
2) You are not constantly looking back to see if your young one has gotten lost in the crowd
3) The child will be rested and can even fall asleep
4) You can hang purses, water bottles, shop purchases, etc off the back (I suggest buying some plastic hooks in advance so you can have some additional hook points on the stroller handle)
I happily rented Disney strollers for years but last year we switched to Orlando
Stroller Rentals. They deliver a stroller to your hotel bell-hop, ready when you check in. Advantages of this:
1) You can bring the stroller out of the park so you can stroll the child all the way to the bus and again from the bus to the room. You can board the monorail with the child still in the stroller.
2) I recall that they are cheaper then Disney strollers
3) They look more comfortable
I suggest personalizing your stroller so it's easy to pick out in the parking areas and to reduce the change of someone deciding to accidentally borrow your stroller. For example tie a bright shirt around the handle, or keep a balloon tied to it.