Someone here at the Monastery got me onto Mime::Lite which I like over Mail::Sendmail which I was using. Mime::Lite makes adding attachments so much easier.

use MIME::Lite;

for (@email_recipients) {

   #-- create the multipart container
   my $msg = MIME::Lite->new (
      From      => $query->param('fromwho'),
      To          => $_,
      Subject  => $query->param('subject'),
      Type       =>'multipart/mixed'
   ) or die "Error creating multipart container: $!\n";
    
   #-- add the text message part
   $msg->attach (
      Type  => 'TEXT',
      Data   => $query->param('message'),
   ) or die "Error adding the text message part: $!\n";
    
   #-- add the ZIP file
   $msg->attach (
      Type     => 'AUTO',
      Path      => "../".$file,
      Filename => $file,
      Disposition => 'attachment'
   ) or die "Error adding $file: $!\n";
                            
   $msg->send;
        
   sleep(1);
}
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Note the use of the delay at the end of the loop. This avoids the batch e-mail appearance to some spam filters. However, it may be moot because many ISPs limit the number of emails you can send at once or in an hour.