If you’re admin panel is password protected by htaccess then plugin will not able to access wp-ajax.php page, which is essential to pull search result from database. Very few sites occasionally uses htaccess rules to protect their admin panel access. But, unfortunately ajax call will not work until you remove htaccess code.
Few references:
1. http://docs.easydigitaldownloads.com/article/201-admin-ajax-blocked
2. https://wordpress.org/support/topic/live-search-only-works-when-logged-in-as-admin