We are at WDW now, also doing it for the first time with no stroller. We have done a variety of things:
1. Put main bag in a day-rental locker and fetched stuff only if needed - like fleeces for chilly late nights / when clothes are wet from wet rides or rain.
2. Just taken small camera plus little cash and one credit card, all in those side pockets you get on shorts. We have a supply of ziplok sandwich bags to keep everything dry (tickets for FPs, camera, phone) - rain, wet rides and sweat get most things damp otherwise!
3. Carried a bag with the barest minimum - but this does tend to fill up and get heavy!
I have thought of getting a water-holster or lanyard for each person to keep their own, water bottle as water for 6 people s the main weight!
It has crossed my mind to just buy a cheap stroller though, but my husband wouldn't do it!