回答:
どうやらそれは次のように簡単です:
add_action( 'admin_init', 'posts_order_wpse_91866' );
function posts_order_wpse_91866()
{
add_post_type_support( 'post', 'page-attributes' );
}
そして、クエリを実行します:
$order_posts = new WP_Query(array(
'post_type' => 'post',
'post_status' => 'publish',
'orderby' => 'menu_order',
'order' => 'ASC',
) );