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__
[download]

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