vnstat can return error in mails which are sent to root mailbox:

Error: Unable to write database "/var/lib/vnstat/eth0"

The solution can be to edit /etc/cron.d/vnstat and replacing "nobody" by "root"

vi
/etc/cron.d/vnstat
if emails dont stop, may help this command "service crond restart"

If one want to delete too mails (visible when one type "root" command in linux), it can be done by removing file
/var/spool/mail/root

rm /var/spool/mail