Is it possible?
Not a good idea, but here's a few things to try:
Usually sites don't want you to have an invisible avatar, so the always do counter measurements, they may have a minimum image size and simply ignore transparency from your image.
Which leaves us with the last option only, that is not useful either, because the website has themes (I use dark theme for instance) depending on your theme your avatar may be invisible to some people but not others, so it may become just a blank square in the end, not very interesting.