** With flash, most times it can be used here using the gigya shortcode. Panos covers the gigya shortcode quite well in the link below.
The flash would have to be hosted elsewhere on the web since it cannot be uploaded here. Also, this would require that the header actually be put into a text widget, and then the text widget resized and repositioned to the header area using inline CSS. This gets kind of complex, and I've never tried it.
Since this is a multi-user platform we all share the same underlying theme and wordpress files and any bad code, or security issue introduced by a user could bring down a bunch of blogs.
The other alternative is an animated gif. The more frames in the animation, the bigger the file, so it is best to keep it simple. Also, the file has to be sized to the exact dimensions of the header. If it is bigger or smaller wordpress is going to want to resize and crop it to fit, and that will kill the animated gif.