Hi Curiousgeorge,
Boy, that's a tough one. I have seen child harnesses for sale at some larger chain stores like
WalMart, Target, Shopko, etc... and there used to be a type of leash which you put around the child's wrist, but I think those were recalled because the coloring on the leash came off on children's wrists, and the coloring was toxic.
I know where you're coming from because we have a 17 month old little boy who loves to run around, but is too young to know that he should stay close to mom and dad. In other words, if we were to put him down, he would be off like a rocket!

We don't use any type of leash or harness though. We just look for a quiet, grassy little area to let him run around in for 15 minutes or so while we all take a break from the rides and enjoy an icecream treat, and then back into the stroller for more rides and sights. You'll be surprised because there's so much to see, the kids probably won't even get restless in the stroller. They're too busy watching everything!
I wouldn't worry too much about your four year old wandering off unless he has a tendancy to still to that. However, the two year old is a little different story. Sometimes in unfamiliar areas when there are a lot of people around, kids have a tendancy to stick closer to mom or dad. But I've seen some kids just run off regardless of their surroundings. You know your kids best and what to expect.
If you opt to use a child leash or harness though, I wouldn't let anyone make you feel bad. Just say, "Hey, I love and care about my kids so much that I don't want to lose them!"
As for the stroller, I would take my own because once you leave the front gates of the DL or DCA, which is where you have to return rental strollers, you still may wish you had a stoller to use between trips from your hotel to DL/DCA and back again, and strollers are also handy at airports. Plus, with your own double stoller, you always have somewhere to strap in the kids so they don't wander off.
Good luck with this.