That's not the email I got, that doesn't seem correct at all. That's a customer service contact email, but not an order email. Maybe their order system got overloaded.
Do you see a charge somewhere? If you go back to their website and try to order that same domain, does it say it is available?
As noted, their system is all automated, as soon as I pressed to submit my order and I paid, I got an email which says "Sign-Up Email" automated with an order number.
Did you check your spam folders? They send like 7 emails total so many of the emails go to spam folder.