Stroller rental prices at Disney are insane. You're better off buying one at Wal-Mart and throwing it out or selling it for pennies (or donate it) when you're done.
I have seen many older kids, including early teens, sitting in the bigger strollers. Personally, I don't care who looks at us funny, I'd rather have happy and not tired kids than cranky, tired messes. Last summer, we had a 16 day trip. At the time, my oldest was 11 and my twins were days away from turning 8. My oldest walked, but we made the decision that the twins would melt down if they had to walk day in and day out. We still had their double wide stroller from when they were babies...and our previous trip to WDW. We brought it and they rode in it 85% of the time. It made it easy for us to keep tabs on them and not have try to herd them where we want to go. Plus, they didn't wear out in the heat. And when one of them walked, my 11 year old sat down a handful of times. After that, our trips were all extended weekend trips. For those 3 and 4 day trips, they walked. We brought a single stroller, which we use as a gear stroller to carry our cooler bag for food. We're heading down again this week. Plan is the same. Everyone will walk, and we'll have the gear stroller. But if one of them gets really tired, I have no qualms with them being pushed a little.
Side bonus...pushing around near 8 year old twins, plus a bag of food hanging off the back, is one HECK of a workout.