I have a Graco IPO stroller that is an umbrella stroller that holds up to 50 pounds. Its lightweight (17 pounds), folds up, but fully reclines and has a little basket. For my 3 year old, this is wonderful at the parks and not too much hassle on the buses. It was $80, I think, so less than the cost of renting one. There is also a Combi Flare umbrella stroller that goes up to 55 pounds and is even lighter (11 pounds) to carry, but I read some negative reviews about the wheels not working properly so went with the slightly sturdier but heavier Graco, and I'm pleased so far.
Joggers will carry more weight, but if you're staying on property...well, I've done the jogger on the bus before and it is NO FUN. I would not personally do that again, though I see plenty of people willing to cart one of them around.
To be fair to the OP, I will say that sometimes, you cannot always predict when kids will fall asleep, get sick, or need a break. For me, our schedule is already off at Disney due to the time change and getting up earlier than usual, so I've definitely had my daughter fall fast asleep at an unexpected time before (like an hour after she got up from her nap at the hotel). It happens. Having toted a deadweight sleeping child from the bus to the hotel room, I appreciate the need for a stroller!