Zen Cart does send the email. If it was encountering an error, it would give you a blank white screen instead of completing the order process.
Or, if you've got an invalid email address specified in the module's settings, then of course it can't send the email. If someone else's email address is in there, then of course you won't get the email.
So ... Zen Cart is sending the email. But the mailserver operated by your hosting company is apparently rejecting the message. Best to contact them for assistance in troubleshooting.
Better yet, use a live gateway service, since with email NOT being a secure communications vehicle, sending cc digits over email is NOT a PCI compliant activity, and your cc company may not be very happy with you using that method.
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