I have an avatar that is connected with my shop name (William, the recycled jeans bear, from William's Giftshop).
I use it all over the internet, wherever I am active. And I use it on my business cards, flyers, banners, etc.
Many people recognize it and are reminded of my work and my shop when they see it, so it is perfect for branding.
I don't use a picture of myself, because I don't want to publish my face online. All you can find from me online are my creations and a lot of words. But never my face. That is how I like it.
My shop isn't about me, but about the things I create.
In general, my presence on the internet is not about how I look, but about what I do.
(If I could have done it all over again, before my work and William's face were so well known, I would have chosen a different shop name (Made by BeaG) and a different avatar, because some people think I sell bears, but it still wouldn't be my face.)