It looks like the other fields are required as part of the form submission, even if they are left blank. Changing the my $req line to the following worked for me:

my $req = POST 'http://www.usps.com/cgi-bin/zip4/zip4inq2',
          [
            'Firm' => '',
            'Urbanization' => '',
            'Delivery Address' => '3601 Fallstone Court',
            'City' => '',
            'State' => '',
            'Zip Code' => '20744',
            'Submit' => 'Process',
          ];
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