You'll have to manually re-query the data from the database, as it's not extracted for that page's output since it's not normally used there.
/includes/modules/pages/checkout_success/header_php.php
Insert the following at the end of the file, before the closing ?>
Code:
$sql = "SELECT customers_firstname, customers_lastname, customers_email_address FROM " . TABLE_CUSTOMERS . "
WHERE customers_id = :customersID:";
$sql = $db->bindVars($sql, ':customersID:', $_SESSION['customer_id'], 'integer');
$cust = $db->Execute($sql);
Then in your template file, you can reference the info like this:
$cust->fields['customers_firstname']
$cust->fields['customers_lastname']
$cust->fields['customers_email_address']