Find custom post type url by author

Query for post type listing_type and author ID within your loop:

$args = array(
    'post_type' => 'listing_type',
    'author' => get_the_author_meta( 'ID' )
);
$listing_post = new WP_Query( $args );
if( $listing_post->have_posts() ){
    while( $listing_post->have_posts() ){
        $listing_post->the_post();
        the_permalink();
    }
}
// reset $post global to original value
wp_reset_postdata();