adding gravatar

below i return the avatar and email (because that is what i need)
but you can make the email optional

var $email;       // String
var $avatar;     // String

function JSON_API_Author($id = NULL)
{
	...
	$this->set_value('email', 'user_email');
	$this->avatar = $this->getAvatar();

	$this->set_author_meta();
}

function getAvatar()
{
	$avatar = get_avatar($this->email);
	preg_match("/src=['\"](.*?)['\"]/i", $avatar, $matches);

        // i return only the link, but you can return the array
	return isset($matches[1]) ? $matches[1] : '';
}

https://ww.wp.xz.cn/plugins/json-api/