No, this isn’t possible. You can grant a custom badge through the API though. See Grant a custom badge through the API for details about that.
You can create a group that adds a group flair to its members’ avatars. This is how the Discourse Team flair is displayed on this forum. To automatically grant a badge, instead of an avatar flair, to members of a specific group, you could either write a badge query, or you could grant the badge through the API.