It sounds like the form is posting directly to their system. As you'll never be able to get control of the return XML. What you'll need to do is have the form post to your server, then you need to construct your own web request as a form post that your server will make to their server. Then your server will receive the response and process the returning XML as needed.
Look into the System.Net.HttpWebRequest class. I have only used this a little, but with it you should be able to construct a complete web request, building the form elements that you received from the user's form post, point it to their page, and a class method will return the response stream to you so you can grab the XML. It's going to turn your server into the browser from the viewpoint of your vendor's site.