#!/usr/bin/perl

use strict;
use HTTP::Request::Common;
use LWP::UserAgent;
use HTTP::Cookies;
use URI::Escape;

my $url = q{http://cgmix.uscg.mil/PSIX/VesselSearch.aspx};

my $ua = LWP::UserAgent->new();
$ua->cookie_jar( HTTP::Cookies->new(
    'file' => '/home/matt/cookies.txt',
));
my $safe_key = uri_escape($key);
my $req = POST $url,
Content_Type => 'form_data',
Content => [
        VesselName => 'FLINTHORN',
        VesselNumber => '',
        VesselCallSign => '',
        VesselHull => '',
        VesselYear => '',
        CheckBoxOutService => '',
        __VIEWSTATE => $safe_key,    ];
my $res = $ua->request($req);print $res->content();
[download]

$VAR1 = bless( {
                 '_content' => 'VesselName=FLINTHORN&VesselNumber=&Ves
+selCallSign=&VesselHull=&VesselYear=&CheckBoxOutService=&__VIEWSTATE=
+SHORTKEYTOSAVESPACE',
                 '_uri' => bless( do{\(my $o = 'http://cgmix.uscg.mil/
+PSIX/VesselSearch.aspx')}, 'URI::http' ),
                 '_headers' => bless( {
                                        'user-agent' => 'libwww-perl/5
+.65',
                                        'content-type' => 'form_data',
                                        'cookie' => '$Version=2; ASP.N
+ET_SessionId=xmtfq5nrirypi245hpi20e45; $Port="80"',
                                        'content-length' => 126
                                      }, 'HTTP::Headers' ),
                 '_method' => 'POST'
               }, 'HTTP::Request' );
[download]

This code works for me:

#!/usr/bin/perl

use strict;
use HTTP::Request::Common;
use LWP::UserAgent;


my $ua = LWP::UserAgent->new();

my $url = q{http://cgmix.uscg.mil/PSIX/VesselSearch.aspx};

my $req = GET $url;
my $res = $ua->request($req);
my ($key) = $res->content() =~ m{name="__VIEWSTATE" value="([^"]+)"};

$req = POST $url,
    Content_Type => 'application/x-www-form-urlencoded',
    Content => [
        VesselName => 'FLINTHORN',
        VesselNumber => '',
        VesselCallSign => '',
        VesselHull => '',
    DropDownListFlag => 'ALL',
    DropDownListService => 'ALL',
        VesselYear => '',
        CheckBoxOutService => '',
    Submit1 => 'Search',
        __VIEWSTATE => $key,
    ];

$res = $ua->request($req);
print $res->content();

__END__
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As you can see Content_Type => 'form_data' is one of mistakes

Thanks!!

Also look to WWW::Mechanize for such things.