Honestly because of your kids ages I would recommend you bring your own stroller. Reason #1 After a long day at the park who's going to carry either sleeping child all the way back to the room from the bus? Reason #2 Who's going to carry them while ou are standing in line waiting for the bus? Reason#3 You definitely need a stroller at the park, but if it's their first time at Disney they might be a little nervous in someone else's stroller.
Honestly I'd bring two seperate strollers - one for each - and not a double stroller. This way you and DH can split up easier and also umbrella strollers are easier to fold and unfold for the bus.
Our kids are 6&9 so we rent a double stroller now...yes we still get the stroller for those ages because they still get exhausted and fall asleep midday. However I always brought strollers for years or at least until the youngest one was 4 yo because someone would fall asleep waiting for the bus or we'd have to carry from the bus to the room.
If you decide not to bring your stroller, make sure you rent the Disney stroller and pay for the ENTIRE LENGTH OF YOUR NEED of the stroller INADVANCE!!! This way you wont have to stand in any line to get a stroller. You can walk up to the attendant get your ticket stamped and get your stroller. Also you can use that same ticket on the same day for all of the parks so you don't have to pay again at each park.