You know how you can make it to where only certain groups can use Signiture...
is there a way to do the same with avatar?

Hello,

As far as I know, this is not possible. You can, however, disable certain users from displaying avatars if you wish. You can do this by going to their profile in the administration panel.

I do not see the option

Under "Special admin only fields" do you see "Can display avatar" ? Choosing no will revoke their ability to use an avatar.