I am in WDW right now and do it every day, did it last year and in 2012. You'll have to make your own personal judgment call on whether you want the younger one riding in the stroller or not. People will have strong opinions both ways. My twins are now 8, and we're a very active family. All 3 of my kids play sports and my wife and I are into bodybuilding. My son plays football and lacrosse and his twin sister is a lacrosse player. Having said that, WDW is a HUGE place and we do tons of walking every day. It's hot, darn hot. We're here for 17 days. When we did our 3 or 4 day quick trips, we didn't bring the stroller. But reality is that on a trip this long, they're going to wear out. In fact, as I type this, my twins are taking a nap in their room (we stay off property). I'd much rather have people look at me funny and have happy kids than force them to walk and have them run out of gas and be miserable. Fact is I see plenty of sub-10 year old kids in strollers here. So for me, yes, we use a double stroller.
Strongly recommend getting a jogging stroller with a pivoting front wheel. Last year we had a double umbrella stroller with smaller front wheels. It was darn heavy and in spots was a major pain (the trolley tracks in Main St in MK). This year, we have a double jogging stroller with pivoting wheel. It's SOOOOO much better. I'm pushing something like 160-170lbs of weight, and it's really easy. Only downside is that it doesn't fold compact, so it's a pain to get on/off the tram.
I also would never rent from Disney. Buy one used and either donate it or sell it when you leave. I'm not even sure what the WDW rental prices are, but I think it's something like $20/day for a double. I bought our stroller from Craigslist here in Orlando. Picked it up the day we arrived, paid $70 for it. I plan to use it for our two weeks here and then either resell it or try to find a family coming to WDW who could really use it and "pay it forward".