ầm : loud, noisy

ãm : to carry in the arms

ấm chè : teapot

ám số : password, code

should be

ám số : password, code

ấm chè : teapot

ầm : loud, noisy

ãm : to carry in the arms

Correction: should be

ám số : password, code

ãm : to carry in the arms

ấm chè : teapot

ầm : loud, noisy

I think that getting Unicode::Collate to work would be the best approach, but here's a hand-rolled one that seems to work the way you want it:

use utf8;
use 5.014;
use warnings;
use List::Util qw/min/;
binmode STDOUT, ':encoding(UTF-8)';

my %order;
{
    my $source = join '', 'aáàảãạăaáàảãạăắ',
                 'ằẳẵặâấầẩẫậbcdđeéèẻẽẹêếềểễ',
                 'ệfghiíìỉĩịjklmnoóòỏõọôốồổ',
                 'ỗộơớờởỡợpqrstuúùủũụưứừửữự',
                 'vwxyýỳỷỹỵz';
    my $cnt = 0;
    $order{$_} = ++$cnt for split //, $source;
    sub vcmp($$) {
        my ($a, $b) = @_;
        for (0..min(length($a), length($b))) {
            my $cmp = ($order{substr $a, $_, 1} // 0)
                      <=> ($order{ substr $b, $_, 1 } // 0);
            return $cmp if $cmp != 0;
        }
        return length($a) <=> length($b);
    }
}

say for sort { vcmp($a, $b) } ('ầm', 'ãm', 'ấm chè', 'ám số');

Perl 6 - the future is here, just unevenly distributed

Accidental duplicate. Please see Re^4: Sorting Vietnamese text instead.