You keep overwriting the hash in your loop. Also, you likely want a hash-of-arrays (perldsc) since you have multiple domains per customer:

use warnings;
use strict;

my %customer_domain_hash;
while (my $line = <DATA>) {
    chomp $line;
    my ($k, $v) = split(/@@@/, $line);
    push @{ $customer_domain_hash{$k} }, $v;
}
use Data::Dumper; $Data::Dumper::Sortkeys=1; print Dumper(\%customer_d
+omain_hash);


__DATA__
CustomerA@@@DomainA
CustomerA@@@DomainB
CustomerA@@@DomainC
CustomerA@@@DomainD
CustomerA@@@DomainE
CustomerB@@@DomainA
CustomerB@@@DomainB
CustomerB@@@DomainC
CustomerB@@@DomainD
CustomerB@@@DomainE
[download]

Outputs:

$VAR1 = {
          'CustomerA' => [
                           'DomainA',
                           'DomainB',
                           'DomainC',
                           'DomainD',
                           'DomainE'
                         ],
          'CustomerB' => [
                           'DomainA',
                           'DomainB',
                           'DomainC',
                           'DomainD',
                           'DomainE'
                         ]
        };
[download]