Remove drupal parameters from form submitted via POST
You can make a call to external website using drupal_http_request in form submit handler.. Define extra submit handler using hook_form_alter and 

$form['#submit'][] = 'YOUR_FORM_SUBMIT_HANDLER';


For Example 

// Adds a submit handler/function for your form

function YOUR_FORM_SUBMIT_HANDLER($form, &amp;$form_state) {

    $options = array();
    // Array keys are matching the key that the remote site accepts. URL encoding will be taken care later.
    $options['data'] = array(
        'firstName' =&gt; $form_state['values']['firstName'],
        'lastName' =&gt; $form_state['values']['lastName'],
        'email' =&gt; $form_state['values']['email'],
        'name' =&gt; $form_state['values']['name'],
        'phone' =&gt; $form_state['values']['phone'],
    );
    $options['data'] = http_build_query($options['data']);
    $options['method'] =&gt; 'POST';
    $url = 'https://[url path to external site]'; 

    // Put your additional headers here. Cookie can be set as well. 
    $headers = array('Content-Type' =&gt; 'application/x-www-form-urlencoded');

    $options['headers'] =&gt; $headers;

    // Submits data to the remote server from the drupal server. User will remain in the Drupal form submission chain.
    $response = drupal_http_request($url, $options);

}