Just a note - the City Mini that apple stroller rents out is not actually a jogging stroller. It's made by the "Baby Jogger" company, but it's not suitable for jogging.
That said, the City Mini is a great stroller for Disney. If i were you I'd either rent a City Mini or CIty Elite (depending on your kids' heights/weights) and then rent a glider board to make it a sit and stand stroller. Orlando Stroller Rental has this option - I'm not sure about other rental companies.