Your customer HAS* received a confirmation email. The following steps will help your customer find that email.
1) Make sure they are checking the correct email account. The email will go to whichever email account they used to open the account. Occasionally, customers will use (what they consider) a secondary email account to order from you.
2) Suggest they check their Spam/Junk email. We do everything we can to send the email to their inbox, but we cannot guarantee 100% of FIRST confirmation emails arrive in their inbox. Any emails that arrive in their Spam folder should be manually rerouted to their Inbox. Following that, any future orders should arrive comfortably in their Inbox.
* Occasionally, customers use fake email addresses. We're not happy about that, but it's unavoidable. In that case, they will definitely NOT get the confirmation email.