Have you looked at http://learn.elgg.org/en/latest/guides/web-services.html already?

In the default web services there is no such function. There is plugin which is unfortunately a bit old with a lot of deprecated functions but it still works https://github.com/markharding/elgg-web-services

You could use the http://yourwebsite.com/services/api/rest/xml?method=user.get_user_by_email

But I have no idea how to pass the email parameter that is required. I modified that function so it returns the username of the logged in user, but I think that is not what you are looking for.

I think that passing parameters is broken in elgg web_services, but I am not sure.

I tried adding  ?email=addres@test.com

which is usually the method on how to pass parameters.

No, it is not broken. Passing additional parameters must be done with &

So if you add the function below you can get a username with the following url:

http://yourwebsite.com/services/api/rest/xml?method=user.get_user_by_username&username=test

/**
 * Web service to get a user registered with an user ID
 *
 * @param string $username to check for
 *
 * @return string $founduser the name of the user
 */
function user_get_user_by_username($username) {
        if(!$username){
                $user = elgg_get_logged_in_user_entity();
        } else {
                $user = get_user_by_username($username);
        }

        if (!$user) {
                throw new InvalidParameterException('registration:usernamenotvalid');
        }

        $founduser = $user->username;

        return $founduser;
}

expose_function('user.get_user_by_username',
                                "user_get_user_by_username",
                                array('username' => array ('type' => 'string', 'required' => false)
                                        ),
                                "Get Username by username",
                                'GET',
                                false,
                                false);

But to be honest, this function seems a bit silly.