How to link to the image editor’s Edit Image function?

You may add the following scripts:

wp_enqueue_script( 'wp-ajax-response' );
wp_enqueue_script('image-edit');
wp_enqueue_style('imgareaselect');

Maybe this already helps?

I think the html in the front-end should be similar to the html in /wp-admin/media.php?attachment_id=...&action=edit