Update: changed the client code and results to use the correct port (reversing a change I made while testing).

In your first client, you need to flush your output, otherwise nothing is sent to the server, and, as jethro said, you need two linefeeds between the headers and body of your request. You can try the following:

#!/usr/bin/perl
use strict;
use Socket;
use IO::Handle;

# initialize host and port
  my $host = 'localhost';
  my $port = 8999;
#  my $port = 80;
  my $proto = getprotobyname('tcp');

# get the port address
  my $iaddr = inet_aton($host);
  my $paddr = sockaddr_in($port, $iaddr);
  # create the socket, connect to the port
  socket(SOCKET, PF_INET, SOCK_STREAM, $proto) or die "socket: $!";
  connect(SOCKET, $paddr) or die "connect: $!";

  SOCKET->autoflush(1);

# create data
  my $data = <<"START" ;
<?xml version="1.0"?>
<methodCall>
<methodName>sum</methodName>
<params>
<param>
  <value><double>1.12</double></value>
</param>
<param>
  <value><double>4.12</double></value>
</param>
</params>
</methodCall>
START
  my $header = <<"START" ;
POST /quotes.cgi HTTP/1.0
Host: localhost
Content-Type: text/xml
START

$header .= "Content-length: " .  length($data) ;
my $content = $header. "\n\n" . $data ;
print $content ;

print SOCKET "$content\n" or die "print SOCKET: $!";

print "waiting for response\n";

my $line;
while ($line = <SOCKET> )
{
  print $line;
}
close SOCKET or die "close: $!";
[download]

On my system, this produces:

POST /quotes.cgi HTTP/1.0
Host: localhost
Content-Type: text/xml
Content-length: 209

<?xml version="1.0"?>
<methodCall>
<methodName>sum</methodName>
<params>
<param>
  <value><double>1.12</double></value>
</param>
<param>
  <value><double>4.12</double></value>
</param>
</params>
</methodCall>
waiting for response
HTTP/1.1 403 Forbidden
Date: Mon, 15 Dec 2008 16:09:28 GMT
Server: libwww-perl-daemon/5.818
Content-Type: text/html
Content-Length: 53

<title>403 Forbidden</title>
<h1>403 Forbidden</h1>
[download]

I am not familiar with the Frontier XML RPC2 protocol, but looking at the server it appears your request must use the path /RPC2. From Frontier/Daemon.pm:

    while ($conn = $self->accept) {
        my $rq = $conn->get_request;
        if ($rq) {
            if ($rq->method eq 'POST' && $rq->url->path eq '/RPC2') {
                ${*$self}{'response'}->content(${*$self}{'decode'}->se
+rve($rq->content, ${*$self}{'methods'}));
                $conn->send_response(${*$self}{'response'});
            } else {
                print STDERR "got an invalid request\n";
                $conn->send_error(RC_FORBIDDEN);
            }
        }
        $conn->close;
        $conn = undef;          # close connection
    }
[download]

Update: Change the path in your request to /RPC2 and it appears to work:

#!/usr/bin/perl
use strict;
use Socket;
use IO::Handle;

# initialize host and port
  my $host = 'localhost';
  my $port = 8999;
#  my $port = 80;
  my $proto = getprotobyname('tcp');

# get the port address
  my $iaddr = inet_aton($host);
  my $paddr = sockaddr_in($port, $iaddr);
  # create the socket, connect to the port
  socket(SOCKET, PF_INET, SOCK_STREAM, $proto) or die "socket: $!";
  connect(SOCKET, $paddr) or die "connect: $!";

  SOCKET->autoflush(1);

# create data
  my $data = <<"START" ;
<?xml version="1.0"?>
<methodCall>
<methodName>sum</methodName>
<params>
<param>
  <value><double>1.12</double></value>
</param>
<param>
  <value><double>4.12</double></value>
</param>
</params>
</methodCall>
START
  my $header = <<"START" ;
POST /RPC2 HTTP/1.0
Host: localhost
Content-Type: text/xml
START

$header .= "Content-length: " .  length($data) ;
my $content = $header. "\n\n" . $data ;
print $content ;

print SOCKET "$content\n" or die "print SOCKET: $!";

print "waiting for response\n";

my $line;
while ($line = <SOCKET> )
{
  print $line;
}
close SOCKET or die "close: $!";
[download]

POST /RPC2 HTTP/1.0
Host: localhost
Content-Type: text/xml
Content-length: 209

<?xml version="1.0"?>
<methodCall>
<methodName>sum</methodName>
<params>
<param>
  <value><double>1.12</double></value>
</param>
<param>
  <value><double>4.12</double></value>
</param>
</params>
</methodCall>
waiting for response
HTTP/1.1 200 OK
Date: Mon, 15 Dec 2008 16:17:46 GMT
Server: libwww-perl-daemon/5.818
Content-Length: 128
Content-Type: text/xml

<?xml version="1.0"?>
<methodResponse>
<params>
<param><value><double>5.24</double></value></param>
</params>
</methodResponse>
[download]