Rhapsody

Thanks to @datakick the answer is fairly simple.  The customization field id is textField0 which was determined using the Chrome Inspect option.
The url for the product page has ?subdmn=desired_subdomain appended at the end where desired_subdomain is the text to be populated in the customized field box.
I used a Google search to get the correct Javascript syntax and inserted the script below on the Custom Code page in the section labeled Add extra JavaScript to your pages.
window.onload = function(){ var subdmn = (new URLSearchParams(window.location.search)).get("subdmn"); document.getElementById("textField0").value = subdmn; };  
This concept can be used for virtually any shop and product with customized fields.  Just change textField0 to whatever the ID is that matches the customized field you want to populate

I am so embarrassed!  I had overrides created on a much older version of ThirtyBees across all three shops.  Once I updated the overrides using the newer core code, the problem was fixed.
My apologies for wasting your time.  Thanks for all the troubleshooting assistance!

Problem solved! 
It is something with the PHP mailer.  I have changed outgoing mail to use SMTP and the problem is solved.  I can't explain why test messages were sent ok using the PHP mailer but actual shop messages are not. I have been using the PHP mailer successfully up until now.
ps - a big thank you and Happy New Year to all who provided troubleshooting suggestions.