here are some of my understandings:
"My application's inventory system keeps track of all the products", this is about the local inventory , which manages all the products in stock. so this local inventory is used for "check availability of products".
we will only need to send JMS order message to Manufacturer's inventory system when number of the product in local inventory reach below certain threshold, which could be done either manually through separate jsp
or automatically through some daemon or other mechanism. I don't think manufacturer's system will send update message to client's system .