So ... since the email is showing in the Archive, that means Zen Cart successfully prepared and sent the email, successfully delivering it to the mailserver engine for routing.
Whatever happens after that is up to the mailservers, and is outside the control of Zen Cart.
As explained in the following articles, sometimes switching to "sendmail -f" or SMTPAUTH may help work around the problem if it's because the server is rejecting the email because it's sent from an unauthenticated PHP script.
http://www.zen-cart.com/wiki/index.php/Troubleshoot_-_Understanding_Causes_Of_Lost_Emails
http://www.zen-cart.com/wiki/index.php/Troubleshoot_-_Email_Problems
Additionally, it's not uncommon for internal mail routing problems to occur within a network. And, since your server could conceivably be sending from itself ... back to itself ... it may very well be dumping the messages into a black hole because it sees that as spammy. Using an external destination email address by entering it into the Contact-Us Pulldown (in prescribed format) may help get around that problem.
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