Switch to user Zimbra

su zimbra

Add credentials to relay_password file:

nano /opt/zimbra/conf/relay_password

Add:

smtp.sendgrid.net yourSendGridUsername:yourSendGridPassword

Apply configuration to Zimbra

postmap /opt/zimbra/conf/relay_password
postmap -q smtp.sendgrid.net /opt/zimbra/conf/relay_password
zmprov ms `zmhostname` zimbraMtaSmtpSaslPasswordMaps lmdb:/opt/zimbra/conf/relay_password
zmprov ms `zmhostname` zimbraMtaSmtpSaslAuthEnable yes
zmprov ms `zmhostname` zimbraMtaSmtpCnameOverridesServername no
zmprov ms `zmhostname` zimbraMtaSmtpTlsSecurityLevel may
zmprov ms `zmhostname` zimbraMtaSmtpSaslSecurityOptions noanonymous
zmprov ms `zmhostname` zimbraMtaRelayHost smtp.sendgrid.net:587

Now wait for a couple of minutes and start testing it by sending out some test mails. Follow the log file to see if it works or not:

tail -f /var/log/zimbra.log