We are going on our first trip to WDW in a few weeks but on the advice of many people we rented a double stroller, a Bob dualie. We have a 5-year-old and a 2 1/2-year-old and although our older daughter hasn't ridden in a stroller in ages, from what I hear a day at Disney isn't like anything else! We are planning to stay at the parks all day, not taking breaks during the day, since our younger daughter will nap in the stroller as needed. As energetic as my 5-year-old is I don't feel like dealing with crankiness if we want to get somewhere in a decent amount of time. I'm sure she'll be walking most of the time but it will be invaluable to have it when we need it, and the extra seat will give us a place to put backpacks or whatever when she's not sitting in it. I would get the double for that alone, since you'll probably be bringing a lot with you with both kids.
We are walking nightly, but not at our normal Disney pace. They can easily handle walking 1.75 miles at a time. But I'm not sure they can handle a week of Disney. 
