Yes it is possible to use animated gifs for avatars. But you're going to have an extremely difficult time using the one you posted. Here's why;
1. The animation is 92 frames. At best, you're going to be able to use 20 frames of an animation before you reach the 19.5 KB size limit.
2. The animated file size is 2.2 MB. Even after you re-size the image down to 80x64, it's still in the neighborhood of 330KB. No where near usable.
Now, there are ways to make the file size smaller. You could cut out some frames and make the animation shorter. Or you could also try increasing the loss%. Basically, the image loses quality, but it's a smaller file size.
Your best best, if you really like the image, is to use it as your signature instead. It would be easy to to crop it to a usable size and then flip it horizontally.