您在ajax中遗漏了一些东西。请查找以下更新的代码，这些代码将帮助您解决问题

function add_ajaxurl_cdata_to_front(){ ?>
   <script type="text/javascript"> //<![CDATA[
       ajaxurl = \'<?php echo admin_url( \'admin-ajax.php\' ); ?>\';
   //]]> </script>
<?php }

add_action( \'wp_head\', \'add_ajaxurl_cdata_to_front\', 1);
add_action( \'wp_footer\', \'add_js_to_wp_footer\' );
function add_js_to_wp_footer(){
    ?>
   <script type="text/javascript">

   jQuery(\'document\').ready(function(){
        jQuery(\'#reg_email\').on("change",function(){            
            jQuery.ajax({
                type: \'POST\',
                url: ajaxurl,
                data: {"action": "custome_ajax_email_check"},
                success: function(data){
                    console.log(data);
                    data = jQuery.parseJSON(data);
                    console.log(data.result);
                    if(data.result) {
                        alert(\'Email exists!\');
                    } else {
                        alert(\'Email does not exist!\');

                    }
                }
            });
        });
    });
   </script>
<?php }

add_action( \'wp_ajax_custome_ajax_email_check\', \'custome_ajax_email_check\' );
add_action(\'wp_ajax_nopriv_custome_ajax_email_check\', \'custome_ajax_email_check\');

function custome_ajax_email_check(){
   $email = $_POST[\'email\'];

   // do check
   if ( email_exists($email) ) {
       $response->result = true;
   }
   else {
       $response->result = false;
   }

   echo( json_encode( $response));
   wp_die();
}