I agree that the tripod mount is your best option. I would mount it on a monopod, sections/ legs all closed down, and ended up with some nice results on Big Thunder Mountain. Actually most of the video was shot aimed back at us and my niece (first WDW visit) and ended up with some great reaction shots. A gorillapod would also work or if you look around you can also find handles/grips to attach it to.
Also attached the bike connector kit onto the strollers handle bar and just had it shoot when we were walking around the park. The results were nothing great, but they gave a neat perspective of everything. My niece got a Minnie mouse hat and the way the ears popped up, made some of the video look like Minnie Mouse was giving us a tour!

also had a nice speed up video of our walk from the Swan Dolphin to Epcot. Looking forward to doing the same thing this year and hope the ability to view the video on the iPhone will help me frame the video better.