Look in Postman's Email Log, the exact error that occurs when the Contact Form tries to send an email will be shown there.
If the emails do not show up in the log, then there is a bug in the theme. This code:
require_once( $_SERVER['DOCUMENT_ROOT'] . '/wp-load.php' );
Is definitely suspicious. The developer should not be calling a WordPress core file directly.
Perhaps ignore the theme's Contact Form and use a real one like Contact Form 7?