Hi, Steveny. I think that after you rent a Walt Disney World stroller for the day, you'll be happy to part ways with it!
Seriously though. Dude. I highly recommend bringing your own stroller to Disney World. It pays dividends at the very beginning of your vacation. First, it makes navigating the airport much easier. Second, it makes getting to your room swifter once you get off of the bus. It makes your commute from your room to the park bus much more predictable. Your own stroller likely won't tip over if a feather lands on it while unmanned in the parks. Your kid will likely be way more comfortable in his/her own stroller. You'll have better storage for your stuff. You're more likely to be able to spot your own stroller in a sea of strollers when disembarking from an attraction. You'll have a better place to put your coffee.
If your child is asleep in your personal stroller, the kid can keep sleeping until it's time to board the bus.
In fact, the biggest - and only - difficulty, in my opinion, to bringing your own stroller is collapsing it and carrying it on the bus when you are en route to or from a park. It's 15 minutes of pain but that's nothing compared to the gain.
If you decide to rent a Disney stroller, you return it at the front of the park when you leave. You then pick a new one up when you return to a park. Here's more
information on stroller rentals.