Say you have a simple resource.cgi:
#! /usr/bin/perl
use warnings;
use strict;
use Reader;
use CGI qw/ :standard /;
print header("text/plain"),
      "Contents:\n",
      Reader::data,
      "-" x 40, "\n";
Its output is
Content-Type: text/plain; charset=ISO-8859-1
Contents:
This is a data file
with some very interesting
bits.
----------------------------------------
The fun part is in Reader.pm, which begins with familiar-looking boilerplate:
package Reader;
use warnings;
use strict;
use Fcntl qw/ :DEFAULT :flock :seek /;
use POSIX qw/ setsid /;    
Next it defined rendezvous points:
my $PIDFILE = "/tmp/reader.pid";
my $DATA    = "/tmp/file.dat";
my $PIPE    = "/tmp/reader.pipe";
The sub import is called as part of use Module. If the daemon is already running, then there's nothing to do. Otherwise, we fork off the daemon and write its process ID to $PIDFILE.
sub import {
  return unless my $fh = take_lock();
  my $child = fork;
  die "$0: fork: $!" unless defined $child;
  if ($child) {
    print $fh  "$child\n" or die "$0: write $PIDFILE: $!";
    close $fh             or die "$0: close $PIDFILE: $!";
    return;
  }
  # daemonize
  close $fh;
  chdir "/";
  open STDIN,  "<", "/dev/null";
  open STDOUT, ">", "/dev/null";
  open STDERR, ">", "/dev/null";
  setsid;
  open $fh, "<", $DATA or die;
  undef $/;
  my $data = <$fh>;
  close $fh;
  while (1) {
    open my $fh, ">", $PIPE or die;
    print $fh $data         or die;
    close $fh;
  }
}
Every client needs to wait its turn getting a lock on $PIDFILE. Once we have the lock, we then check that the process identified is still running and create the named pipe if necessary.
sub take_lock {
  sysopen my $fh, $PIDFILE, O_RDWR | O_CREAT or die "$0: open $PIDFILE: $!";
  flock $fh => LOCK_EX                       or die "$0: flock $PIDFILE: $!";
  my $pid = <$fh>;
  if (defined $pid) {
    chomp $pid;
    if (kill 0 => $pid) {
      close $fh;
      return;
    }
  }
  else {
    die "$0: readline $PIDFILE: $!" if $!;
  }
  sysseek  $fh, 0, SEEK_SET or die "$0: sysseek $PIDFILE: $!";
  truncate $fh, 0           or die "$0: truncate $PIDFILE: $!";
  unless (-p $PIPE) {
    system("mknod", $PIPE, "p") == 0
                            or die "$0: mknod exited " . ($? >> 8);
  }
  $fh;
}
Finally, reading the pipe is trivial:
sub data {
  open my $fh, "<", $DATA or die "$0: open $DATA: $!";
  local $/;
  scalar <$fh>;
}
Don't forget to return a true value from the module:
1;
You'll note that operations can still fail in the daemon. For your sanity, you'll want to log events somehow rather than choking silently.
As to whether hosts will permit long-running processes, that will vary from provider to provider, but even if your daemon is killed off from time to time, the code above will restart it on demand.