Programs like ImageReady (which is packaged with Photoshop) will allow you to open video files. From there it's a simple matter of saving to .GIF format (.JPG is not capable of animation).
I did
Sliverthorn's avatar that way. I extracted the file from the MW2 CD, opened it with Bink (or something similar), saved the file to a .AVI (any video file will work I believe), then opened it with ImageReady.
After a little bit of cleaning up some resizing, and a little bit of looping (in a failed attempt to make it seamless) I just had to save it into a .GIF file, upload it, then send it to him so he could set it as his avatar.

I actually did a bunch like this, one was a
Mad Dog (err...
Vulture) for
Gwyd over at MWL, but then they removed the ability to have custom avatars. I had a
Nova (
Blackhawk) for a while, but the seam between the loop was bugging me, so I removed it.

Edit: One thing I forgot to say, is you have to have some sort of way to get access to the video clip in the first place in order to do anything with it. How you do that varies, I've never had to find a way to actually create the footage, so I can't help much there.