I had this problem with my main email account, and finally had to change my main email to my gmail account. I do now get game emails there. Some email companies go too far with their spam filter, and for some reason, ESO email didn't even make it to my spam filter on my main email.
It's certainly worth changing an email address when this happens. It isn't necessarily anything to do with ZOS or ESO, I've had a similar situation with an organisation I belong to where every time they sent an email I failed to receive it. I checked with them and with my ISP and the emails just disappeared somewhere into the ether. Eventually I gave the organisation one of my other email addresses and I haven't missed a single email since.
Have you tried the following when requesting your access code?
Add "noreply@mail.elderscrollsonline.com" to your email account's list of approved e-mail senders.
Check your spam or junk mail folders.
If you are using Gmail with the Inbox categories feature, also check your Social folder.
If you are using a firewall or anti-virus, temporarily disable them before requesting a new code.
If you have not received your access code after trying the steps above, you should contact customer support to request an email address change by selecting "No Longer Have Access / Can't Remember Email on Account" in the Issue Type field on this form.