I am in the same boat you are alad, a solution to this would be awesome. Interesting that local video in iOS with webView works so easy.
Finally I found a solution...
The problem is that AIR on Android stores the application data in the protected directory /data/... which is not accessible by StageWebView (or any browser app).
So you have to copy the video file you want to play to a temporary pubic directory on the sd-card for example and use this in StageWebView.
You also have to create a temporary html-page with the video-tag and load this page in StageWebView (not the string) because external resources are only loaded in StageWebView when using an URL.
On iOS the application storage directory is readable for StageWebView so you don't have to do this step.
Hope this helps
Nice that you`ve figured it out, however i still have problems to make it work.
Can you share some code you`ve developed (html page, loading instructions including paths, and paths of video)