By default, the Vector Watch is displaying all your phone-level notifications, so whatever you see on your phone as notifications will be forwarded to the watch.
If when receiving an e-mail your phone does not vibrate and show the e-mail on the phone locked screen, then they won't be forwarded to the watch.
Please go to the phone's Settings / Notifications / Mail and check that "Allow Notifications" is on, then for each account (assuming there's multiple ones), turn on "Show in notifications centre".
If you'd like to get only the VIP e-mails sent to the watch, see this article.
Another reason why the e-mails might not reach your phone could be the fact that you have filtered them out from the Vector Watch app on your phone.
Open the Vector Watch app on the phone, and go to Settings / Alerts. If "Mirror my phone" is on, all the notifications will be forwarded to your watch.
However, if "Mirror my phone" is off, you should make sure that the Mail app is enabled in the list bellow.