Here is the code with the above suggestions applied.

#!/usr/bin/perl

use strict;
use warnings;

use LWP::UserAgent;
use HTML::LinkExtor;
use URI::URL;

my $site      = shift @ARGV;   # Why not just take an uri as argument?
my $domain    = shift @ARGV;

# Should really be an uri.
# Assume it's a HTTP server if not.
if ($site !~ /^\w+:/) {
   $site = "http://site";
}

my $first_uri = URI->new($site)->canonical;
my @to_visit  = ( $first_uri );
my %seen      = map { $_ => 1 } @to_visit;

my $ua = LWP::UserAgent->new();

while (@to_visit) {
   my $uri = shift(@to_visit);

   # Make the parser
   my $p = HTML::LinkExtor->new(sub {
      my ($tag, %attr) = @_;

      # Only interested in A elements.
      return if $tag ne 'a';

      # Only interested in the HREF attribute.
      return if not exists $attr{href};

      my $link_uri = URI->new_abs($attr{href}, $uri)->canonical;

      # Ignore links outside of the domain.
      return if $link_uri->rel($first_uri) eq $link_uri;

      # Ignore links already in the queue and links already visited.
      return if $seen{$link_uri}++;

      push @to_visit, $link_uri;
   });

   my $response = $ua->request(
        HTTP::Request->new(GET => $uri),
        sub { $p->parse($_[0]) }
   );

   $p->eof;
}

my @links = keys %seen;
[download]

You should use LWP::RobotUA instead of LWP::UserAgent for this kind of application.

Untested.

Updated: The call to request was accidently removed! oops. Readded. Added $p->eof.