If you have a car, then you always have that option over the buses. Sometimes the bus/boat/monorail is fun/best, but sometimes the car is a good option too.
I think it makes sense to avoid renting one at WDW.
If your stroller is heavy, then find an option which is light, easy to maneuver, and is the proper height. If you - or your spouse- is tall- an cheap umbrella stroller will be a thing of misery to push all day. Heavy and bulky is also no fun.
I don't know what you are spending on this trip, but I'd strongly encourage you to consider investing is a higher quality easy fold stroller with very good wheels. "Umbrella stroller" is a highly variable term: could weigh 100 pounds, could be super-flimsy.
Ideally you need: storage, shade cover, recline, good ball bearings, easy folding! you MUST have a RAIN COVER of some kind! (a plastic poncho plus rubber bands can work) You might well be able to borrow one or buy one second hand!
Conversely, the ideal at WDW is not to have a super high end stroller.
Ideally, you'll want to personalize your stroller- tie bright colored ribbons, bright stickers. At every attraction, you'll park it in a lot with 1000 other strollers. (all the same make and style!) While you are on the ride CM's will MOVE it!!!!! It is not enough just to remember where you parked it. It REALLY helps to have some sort of CLEAR visual marker!!
No matter what, at the end of the day, when you are tired, you'll mostly need to be able to fold you stroller and stow it. (either in the bus, or your resort room, or the car) You'll very much appreciate a stroller that is lightweight and easy to fold.
In the parks, you'll constantly have to navigate crowds and stores, and find parking spots to leave your stroller. Having a compact stroller is a little like having a compact car in a busy city. It's just so much easier to find parking, and so much easier to drive in alleyways!