Like Tanktalus mentioned, sound like a job for LWP.

Simply post the required form data to the ServerB via LWP. The script on ServerB will need to send some sort of response which you can process in the script on ServerA

my $ua = LWP::UserAgent->new();
my $response = $ua->post(
  $url,
  {
   'field1'=>$data1,
   'field2'=>$data2,
   etc
  );

if ($response->is_success) {
  my $accepted = $response->content;
  #process data
} else {
  #error
}
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