I'm not sure what you mean with "integration page" - you could use the template of the checkout success page, but I'd rather say "abuse" because such logic does not belong in templates.
Magento has an event system where you can create observers that are executed on certain events. For placed orders there is the event
sales_order_place_after and it receives the order model as "order" parameter. If the customers have to enter their phone number, you can access it through this model:
$order = $observer->getData('order');
$phoneNumber = $order->getCustomer()->getTelephone();
Read more on how to create observers in the devdocs: Events and Observers