Depends on what the request looks like. If you make a GET request, (one where the parameters are transmitted within the URL), you could have it first go to your script, you record it, and then issue a redirect

use CGI; 
(store parameters etc); 
print $cgi->redirect('the.other.host/url-with-parameters');
[download]

I don't think that method will work if you have a POST method in your form.

In that case, you could accept the request in your script, record the parameters, then do a LWP::Simple request in the background, and send the results of that request to the client. Note that in that case, the other server will see ALL the requests as coming from your server's IP - which may not be what you want.

Update:Right, as b10m pointed out, LWP::Simple doesn't do POST, only get (the fact that it accepts the URL as the single parameter should have clued me in). Yes, you'd have to use LWP::UserAgent, and it's post method.