Не могу понять, что мешает объявлению ORDER BY не влиять на результаты запроса ...
SELECT `vendor_orders`.`id`, `vendor_orders`.`vendor_order_id`,
`vendor_orders`.`order_number`, `vendor_orders`.`customer_order`,
`vendor_orders`.`order_date`, `vendor_orders`.`status`, `vendors`.`name` AS
`vendor_name`, COUNT( `vendor_order_items`.`id` ) AS `order_item_count`
FROM (`vendor_orders`, `vendors`, `vendor_order_items`)
WHERE `vendor_orders`.`aid` = 'c4ca4238a0b923820dcc509a6f75849b'
AND `vendor_orders`.`vendor_id` = `vendors`.`vid`
AND `vendor_order_items`.`vendor_order_id` = `vendor_orders`.`vendor_order_id`
GROUP BY `vendor_orders`.`id`
ORDER BY 'order_item_count' DESC
LIMIT 0,10