As far as I know wagons are no longer allowed. Your children are old enough to walk unless they have some sort of disability. I will tell you we went for my DD first WDW trip when she was 4 (almost 5). We debated back and forth about the stroller. I was worried she wouldn't make it walking. So i had the contact info for a stroller rental company ready if we needed it. We ended up doing a 10 day trip 7 WDW, 3 universal and my 4DD walked the whole trip. Of course when we were at the parks late she was dead and would crash as soon as we got on the bus, so there were some nights we were carrying her but that was our fault for trying to keep her up way past her bed time.
Just do your best to minimize back tracking around the park. Create a good plan that flows through the park. I love using touring plans to plot out our days - which FP, attractions or shows, where we will be eating. When you can visualize the park it helps to figure out when you should try and hit certain things.
You are not constantly walking there, plan to hit shows which provide a break, plan some TS meals which will allow for a longer break indoors. You will survive and if you can't call and rent a stoller. Don't get the ones from the park -- they seem so uncomfortable and they won't help you when you have to return them and you are left walking to the bus stop or walking from the bus to the resort room.