I'm just gonna put my comment into an answer because otherwise there won't be one.
It is very unlikely that animated avatars will ever be supported on the Pokebase. There are several reasons for this. As many of you know, the avatars that the Pokebase uses come from Gravatar. The Pokebase avatar on any given account is the current Gravatar avatar of the email attached to that Pokebase account, since each Gravatar account is also attached to an email.
While this remains true (as opposed to PM allowing us to upload our own avatars), you will not be able to use animations as your avatar, since Gravatar does not support them. And if you look it up and it says Gravatar supports GIFs, it's technically true, but it won't be animated. It just takes the first frame of your selected GIF and uses that as your avatar.
Oh yeah, and also I have a decent guess why PM won't let us upload images: It takes storage, which costs money. Why not make other people pay money instead?