It appears it could be an issue with your hosting provider.
I tried a simple email script to see if it sends:
$to = 'firstname.lastname@example.org';
$subject = 'the subject';
$message = 'hello';
$headers = 'From: email@example.com' . "\r\n" .
'Reply-To: firstname.lastname@example.org' . "\r\n" .
'X-Mailer: PHP/' . phpversion();
mail($to, $subject, $message, $headers);
and I'm not getting an email from that either. Another technique that works on some servers is:
mail('email@example.com', 'the subject', 'the message', null,
but that isn't working either.
This points to configuration issues with your hosting. I'd contact them with these two scripts... until one of them works, our email script won't work either.