Photography of any kind is not allowed on Soarin'. In the pre-show, Patrick does not say no flash pictures or videorecording, but he one-up's it by saying to stow all cameras in the underseat compartment. The queue, the ride system, etc. are all fair game for pics, but like a user posted earlier, the video itself is copyrighted and if you are caught doing it, you will be stopped. There are actually cameras watching all 3 of Soarin's carriages.\
Nonetheless, if you want to watch the video, youtube.com has a gazillion videos of it already.