你很接近。您试图使用的挂钩是从函数调用的wp_transition_post_status():

function wp_transition_post_status( $new_status, $old_status, $post ) {
   // 
   // ...
   //
   do_action( "{$new_status}_{$post->post_type}", $post->ID, $post );
}

因此，要在以下情况下执行代码：product 状态更改将函数添加到挂钩publish_product.
传递给函数的第一个参数是$post_id, 不WP_Post 对象
最后一件事，检查功能中的帖子类型on_publish_pending_post 没有必要，因为挂钩用于特定类型（“发布_post“，”发布_product“”。

Try this code:

add_action("publish_product", "on_publish_pending_post", 10, 2);

function on_publish_pending_post($post_id, $post) {

    $auth_id = (int)$post->post_author;
    $user = get_userdata($auth_id);
    if ( $user === false )
        return;
    $to_email = $user->user_email;

    $name = get_the_title($post_id);

    $fromMail = "approve@openvectors.net";
    $subjectMail = "Design Approval Success";
    $content = \'<p>Your Design has been published !!!</p>\';

    $headersMail  = \'\';
    $headersMail .= \'From: \' . $fromMail . "\\r\\n";
    $headersMail .= \'Reply-To: \' . $fromMail . "\\r\\n";
    $headersMail .= \'MIME-Version: 1.0\' . "\\r\\n";
    $headersMail .= \'Content-Type: text/html; charset=ISO-8859-1\' . "\\r\\n";

    // ------ 
    // echo "<br>recipient: " . htmlspecialchars($to_email);
    // echo "<br>sender: " . htmlspecialchars($fromMail);
    // echo "<br>post_title: " . htmlspecialchars($name);
    // exit();
    // ------

    mail($to_email, $subjectMail, $content, $headersMail);
}