I added some more reporting to the error case of your script, mostly

    print $resp->as_string()
[download]

When I look at http://www.usps.com/ncsc/lookups/lookup_zip+4.html, the original webpage, there are many more fields to fill out :

<FORM NAME="Table1FORM" ACTION="/cgi-bin/zip4/zip4inq2" METHOD=POST>
<INPUT TABINDEX="1" ID="Firm" TYPE="text" NAME="Firm" VALUE="" SIZE=30
+ MAXLENGTH=30 >
<INPUT TABINDEX="2" ID="Urbanization" TYPE="text" NAME="Urbanization" 
+VALUE="" SIZE=30 MAXLENGTH=30>
<INPUT TABINDEX="3" ID="Delivery Address" TYPE="text" NAME="Delivery A
+ddress" VALUE="" SIZE=30 MAXLENGTH=40 >
<INPUT TABINDEX="4" ID="City" TYPE="text" NAME="City" VALUE="" SIZE=30
+ MAXLENGTH=30>
<INPUT TABINDEX="5" ID="State" TYPE="text" NAME="State" VALUE="" SIZE=
+2 MAXLENGTH=2 >
<INPUT TABINDEX="6" ID="Zip Code" TYPE="text" NAME="Zip Code" VALUE=""
+ SIZE=10 MAXLENGTH=10 >
<INPUT TABINDEX="7" TYPE="submit" NAME="Submit" VALUE="Process" ID="Fo
+rmsButton2"> 
</FORM>
[download]

I would send all of them, even with empty content, because the CGI might be very badly written and expect all those fields to be present. But the best bet is always to use a dump of that session via a small proxy to see what fields with which content are actually sent.

perl -MHTTP::Daemon -MHTTP::Response -MLWP::Simple -e ' ;    # The  
$d = new HTTP::Daemon and fork and getprint $d->url and exit;#spider
($c = $d->accept())->get_request(); $c->send_response( new   #in the
HTTP::Response(200,$_,$_,qq(Just another Perl hacker\n))); ' #  web
[download]