Oh, you were probably asking for other sites. You could use a gifmaker then upload it to an image hosting site. But if they're already animations you can upload them, go to Imgur or PhotoBucket.
Well, either you steal a gif somewhere and scale it to fit as your avatar, or you could sit down and make your own. Download gimp if you don't have a full licensed Photoshop, draw some frames and stick it together via the animation filter. There are plenty of tutorials on that, if you are interested.