Not sure I understand what you need.. are you building an app, or have a store?
The best way I can think of to accomplish what you are asking is to create a webhook for the orders/create. This will send the order information to the URL you specify then you can extract the customers email address and send it to the 3rd party so that they can send your satisfaction email.
See the webhooks documentation
https://shopify.dev/api/admin/rest/reference/events/webhook
Cheers,
Gary