Do you answer your door wearing only a half-tied bathrobe?
That may have creepily worked for Rodney Dangerfield, but for the rest of us… Look professional when you need to be professional: If you do make it in Voice Acting, sooner or later you will have a website. Sooner or later you will be in a studio.
Use headshots when appropriate to your goals? Eh; its up to you, but people stay connected more with a face to look at, than a logo, avatar, or a puppy pic. (Not cats. Cats are people, and don’t like having to always fill in for you). In my opinion anyway.