search blog posts by author first name and or last name

Actually, when you set author_name, WP_Query will use get_user_by( 'slug', <author name> ) to find a user with the specified “nice name” (user_nicename) value, and if found, then WP_Query sets the author arg to the found user’s ID.

Therefore you can follow the same approach, but you would use get_users() to find users having the specified first and/or last name and use the author (or author__in) arg with WP_Query. E.g.

$users = get_users( array(
    'meta_query'  => array(
        'relation' => 'OR',
        array(
            'key'     => 'first_name',
            'value'   => $request['author_name'],
            'compare' => 'LIKE',
        ),
        array(
            'key'     => 'last_name',
            'value'   => $request['author_name'],
            'compare' => 'LIKE',
        ),
    ),
    'fields'      => 'ID',
) );

if ( ! empty( $users ) ) {
    $query = new WP_Query( array( 'author' => $users[0] ) );
    // or to include all found users..
//  $query = new WP_Query( array( 'author__in' => $users ) );

    // ... your code.
}

Additional Notes

• You don’t have to do the $data = $request->get_params(); because you could just access parameters via direct array access on the WP_REST_Request object, e.g. $request['author_name'] like in my example above.

• You should register author_name using args in the third parameter for register_rest_route(), or at least, check that it’s not empty before calling get_users().

• I know the print_r() in your code is just for testing, but the function will actually echo the output unless if the second parameter is set to true..