Which is to rent, but not from Disney. There are several off-site rental companies in Orlando that rent strollers for considerably less than the parks charge, and you have the advantage of having it for out-of-park situations as well.
These strollers are normal strollers, the sort that are owned by individuals, not institutional models. The services will deliver them to your hotel, though there is usually a surcharge for delivery.
If your DH is adamant about not wanting to bring yours, then perhaps you should try going without, and then if it becomes obvious that your kids are not going to manage without one, then call one of the services to get a stroller. If I were going to go the rental route, I'd go with an outside company; if I'm going to pay that kind of daily rate, I want the thing 24/7. (Of course, I'm the Stroller queen, I have one kid and 8 strollers; so I'm not real likely to rent unless I'm on the other side of the transaction .
))
Do the math. You didn't mention your trip dates, but if it is summer, your kids will probably poop out pretty quickly on walking; they all do in the unfamiliar heat. The walkways are all white concrete, and most WDW park days involve at least 5 miles of walking. Assuming that you manage without a stroller perhaps 25% of your park days, how much money are you going to end up spending? If your DH is grousing because he doesn't want to deal with an awkward stroller on busses, then perhaps for the same amount of money as you would be spending to rent, or less, you could get a more travel-friendly model of stroller, especially if you have any children's resale shops in your area where you could get a used one. You can always sell it after your trip.