The customer receives the Tax Invoice as per the site setup, but there is no additional notification sent to the customer to indicate that you have received the order. As part of the eCommerce setup is to automatically notify you in the event of an order being placed, and the customer is sent the Tax Invoice at this time there is no need to send an additional notification to the customer.
Don't forget as well that we don't want to spam the customer with communications, too many emails from a website has the very real chance of annoying your customers and disuading them from buying from your store again.