If you're saying you want your PHP form to submit this data to a remote host automatically then what you need is to generate another HTTP request to the remote host (basically proxying the data over to the remote host).
However, if what you're saying is that you want the form data populated on your page to be populated automatically on some other remote host's page then there's simply nothing PHP can do to help you with that. You will need access to the remote host and you'd have to make modifications to that code to accept remote requests that will automatically populate the page's form elements.
The proxy request can be as simple as this...
$postdata = http_build_query($_POST); /* do your own validation here */
$opts = array(
'http' => array(
'method' => "POST",
'header' => "Accept-language: en\r\n" .
"Cookie: foo=bar\r\n", /* if necessary */
'content' => $postdata
),
);
$context = stream_context_create($opts);
$file = file_get_contents('http://www.example.com/', false, $context);
And that basically just proxies what was sent to your PHP script to the remote host. See PHP HTTP Stream Context Options for more details.