I wouldn't bring a jogging stroller to WDW. Their just too bulk and irritating to other guests.
We have an older Maclaren double stroller that reclines completely and is lightweight (for a double stroller!!). However, from experience, I would advise using 2 separate umbrella strollers. Not only are they lighter, easier to carry, less bulky on the bus (no hairy eyeballs from fellow guests when carrying these), and easier to manuever through the crowds but also the kids don't have to stay together. This is especially beneficial when one child wants to continue visiting attractions while the other needs a potty/snack break. Dh and I frequently split up in the parks (that didn't quite come out the way I intended!), taking the kids to different attractions. If one our children is really tired or cranky, I will head back to the resort with him/her. You can't do this with a double stroller. Just MHO.