I have the Skip Hop bag and while I LOVE it for going to the mall, zoo, etc, I would not bring mine to WDW. It does cause tipping on umbrella strollers when your child gets out. In order to pull it off the stroller to bring it with you, you would have to detach it from the stroller and grab the bag itself or re-attach the long shoulder strap in order to put it on your shoulder to carry...it would be a real pain.
I use a travelon bag for anything I need to keep on me at all times, and I keep a cheap diaper bag (I use the freebie one I got at the hospital) with diapers, wipes, changing pad - items I wouldn't be overly upset about if someone stole them - hanging off the back of the umbrella stroller.