delia:
and the emails show up as sent in the database archive.
Then that means it's your email server that's dropping them. You might ask the server administrator what the email-server's logs are saying about why it's choosing not to deliver them.
I'm aware that you keep saying "it worked before the upgrade" from v151. Good. That's important to know.
While Zen Cart did upgrade the email code that handles email transmission (PHP Mailer upgraded to a much more modern version, which is much more compatible with modern mail servers), yours is the first I've heard of any sudden problems. In many cases it improved email delivery.
You haven't mentioned what email-transport is being used, or any other settings data. Those options are covered in the articles I linked to above. SMTPAUTH is what I always recommend people use for most reliable delivery.
If your server is requiring something specific, only the server administrator can tell you ....and probably only after reviewing the server configs and logs.