Runs under a dedicated user; if I su to that user I can read the modules in question without issue, and a perl -e'use MyHandler' as that user works fine as well.

If I add a block like this:

use warnings;

BEGIN {
  open my $fh, ">>", "/tmp/x";
  print $fh "@INC\n";

  open my $in, "<", "/usr/lib/perl5/5.18.2/x86_64-linux-thread-multi/D
+igest/SHA.pm";
  while (<$in>) { print $fh $_; }
  close $in;

  close $fh;
}

use Apache2::Const qw(:common);
use Digest::SHA qw(sha256_hex sha1_hex);
[download]

then /tmp/x ends up with:

/etc/apache2/script /srv/www/perl-lib /usr/lib/perl5/site_perl/5.18.2/
+x86_64-linux-thread-multi /usr/lib/perl5/site_perl/5.18.2 /usr/lib/pe
+rl5/vendor_perl/5.18.2/x86_64-linux-thread-multi /usr/lib/perl5/vendo
+r_perl/5.18.2 /usr/lib/perl5/5.18.2/x86_64-linux-thread-multi /usr/li
+b/perl5/5.18.2 /usr/lib/perl5/site_perl . /srv/www
package Digest::SHA;

require 5.003000;

use strict;
use vars qw($VERSION @ISA @EXPORT @EXPORT_OK);
use Fcntl;
use integer;

$VERSION = '5.84_01';

require Exporter;
require DynaLoader;
@ISA = qw(Exporter DynaLoader);
@EXPORT_OK = qw(
        hmac_sha1       hmac_sha1_base64        hmac_sha1_hex
...
[download]

So the script can definitely read the file.