I'm trying to find a regex that will...

You don't need no steenkin' regex:

open my $fh, '<', 'whatever'; # or whatever

my %hash = ( a => 0, e => 1, i => 2, o => 3, u => 4 );

while ( <$fh> ) {
  chomp;
  
  my $copy = $_;
  my @array;

  while ( my $char = lc chop $copy ) {
    no warnings 'uninitialized';
    $array[ $hash{ $char } ] = 1;
  }

  say if ( grep $_, @array ) == 5;
}
[download]

Pretty fast, too, you'll find...

Comments welcome.

Update 2: Oops, just realised that my code doesn't work! Change the contents of the inner while loop to:

Which is what I had originally, before playing with no warnings 'uninitialized';. And then I didn't test properly.

Mea culpa.