According to my dictionary, the verb "ship" can be used without an object, for example "the new car model will ship next month". The phrase "your order will ship tomorrow" or "your order has shipped" is quite common. "To ship" would in this case mean "to leave our premises".
"Your Amazon order has been shipped" would be incorrect until it actually arrives (until the process of being shipped has finished). "To ship + verb" means "to transport an object" and "has been shipped" means that it was transported but isn't being transported anymore.
An order cannot ship itself. But an order can ship.