<?php
        $address_line_1      = $_POST['address_line_1'];
        $address_line_2      = $_POST['address_line_2'];
        $suburb              = $_POST['suburb'];
        $state               = $_POST['state'];
        $post_code           = $_POST['post_code'];
        $country             = $_POST['country'];
        $email               = $_POST['email'];
        $first_name          = $_POST['first_name'];
        $last_name           = $_POST['last_name'];
        $phone_number        = $_POST['phone_number'];
        $progress_identifier = '1';

            // PROCESS THE FORM
            if ('POST' == $_SERVER['REQUEST_METHOD'] && !empty($_POST['action']) && $_POST['action'] == "new_customer")
            {

                $new_customer = array(
                    'post_title'  => wp_rand(100,999).'-'.wp_rand(),
                    'post_author' => $user_ID,
                    'post_status' => 'publish',
                    'post_type'   => 'customer'
                );

                $pid = wp_insert_post($new_customer);

                add_post_meta($pid, 'address_line_1',      $address_line_1,      true);
                add_post_meta($pid, 'address_line_2',      $address_line_2,      true);
                add_post_meta($pid, 'suburb',              $suburb,              true);
                add_post_meta($pid, 'state',               $state,               true);
                add_post_meta($pid, 'post_code',           $post_code,           true);
                add_post_meta($pid, 'country',             $country,             true);
                add_post_meta($pid, 'email',               $email,               true);
                add_post_meta($pid, 'first_name',          $first_name,          true);
                add_post_meta($pid, 'last_name',           $last_name,           true);
                add_post_meta($pid, 'phone_number',        $phone_number,        true);
                add_post_meta($pid, 'address',             $address,             true);
                add_post_meta($pid, 'progress_identifier', $progress_identifier, true);

                $survey_post = array(
                    'post_title' => $pid . '-' . wp_rand(),
                    'post_author' => $user_ID,
                    'post_status' => 'publish',
                    'post_type' => 'survey'
                );

                $survey_id = wp_insert_post($survey_post);

                global $wpdb;
                $wpdb->insert(
                    $wpdb->p2p,
                    array(
                        'p2p_from' => $survey_id,
                        'p2p_to' => $pid,
                        'p2p_type' => 'survey_to_customer'
                    )
                );
                wp_redirect(home_url());
                exit();
            }
    ?>

exit;

function my_function_to_hendle_ajax() {
$address_line_1      = $_POST['address_line_1'];
$address_line_2      = $_POST['address_line_2'];
$suburb              = $_POST['suburb'];
$state               = $_POST['state'];
//....
// no need in if( 'post' .... )
$new_customer = array(
  'post_title'  => wp_rand(100,999).'-'.wp_rand(),
  'post_author' => $user_ID,
  'post_status' => 'publish',
  'post_type'   => 'customer'
);

$pid = wp_insert_post($new_customer);

add_post_meta($pid, 'address_line_1',      $address_line_1,      true);
add_post_meta($pid, 'address_line_2',      $address_line_2,      true);
// ....
wp_redirect(home_url());
exit();
}

<form action="<?php echo admin_url( 'admin-ajax.php' ); ?>" method="POST">
<input type="hidden" name="action" value="new_customer" />
...
</form>