Since the computer you're accessing your offsite server from is on the VPN that has access to the remote system (and assuming this remote system is under your control), you could have the offsite server return some JavaScript that AJAX-pulls the data–in this case, a full HTML page–it needs from the remote system using a JSONP-like strategy.
I have used this technique for some small projects in the past.
From there, you are free to do with the data what you will–modify it, return it as is, or otherwise.
Personally, I would avoid the use of frames if possible. Of course, when strapped for time, nothing is off limits ;)
However, if the remote system IS NOT under your control (and they don't have a kickass CORS cross-access policy), your only option is, as you said in the body of your question, frames. Here is an example:
<!-- If the remote system is NOT under your control, it CANNOT respond with the X-Frame-Options header set to SAMEORIGIN or DENY! -->
<iframe id="inlineframe" src="http://www.randomwebsite.com" frameborder="0" scrolling="auto" width="800" height="300" marginwidth="5" marginheight="5" ></iframe>
//Depending on the JS you put here, you may get some "unsafe Javascript" warnings from certain browsers
~function()
{
var frame = document.getElementById('inlineframe'),
ref = frame.contentWindow ? frame.contentWindow.document : frame.contentDocument;
// ref is now a reference to the document of the content within the iframe
// You can now do your getElementById/getElementByTagName etc. and otherwise
// manipulate the response as you please.
console.log(ref);
}();
A working example can be found here: http://jsfiddle.net/FTudJ/
The contents of the iframe should be the submitted results of the remote page, or whatever the remote server responds with. If you're not looking to modify anything, then you don't even need the Javascript component.
The key will be the URL you specify as the source of the iframe. Now, if the remote system doesn't support GET requests for form submissions (in your example code I see a "POST" method), coupled with the fact that you have no control over it and your offsite system can't access it in any way, your only other options are:
Instead of supplying the form's submission URL directly, simply supply the form's typical access URL to the iframe. From there, use JavaScript to populate the form with the necessary values, and then submit the form within the iframe as if you were the user. From there, you can scrape the results back into JavaScript or simply display them as is.
Find a way to get your offsite server into the VPN network so that it may communicate with the remote system on your behalf. Of course, if that were easy, I guess you would have done that first!
Redirect your users to the remote site–possibly using a frameset + banner frame like Google used to do, let them do what they need to do there, and then tell them to come back when they're finished.
--
I see that you're using PHP. As long as the server that is executing the PHP has access to the VPN that has access to this remote system–all over traditional HTTP–a simple curl call from the server-side should suffice, as you can use AJAX to pass through the results to the client.
These links may be of some assistance: