Well, that's a start ... but
1. not all VPS's are alike (package offerings) - many have a cap on the number of messages one can send (spamming protection for their network) and
2. there are different Linux distros (Ubuntu/CentOS).
CentOS/Ubuntu/Other? VPS - hosting provider? Package (not all provide the same). Hosting providers many times have FAQ's and how-2's ... not all are setup the same. Have you consulted hosting providers forums?
Generic commands seeking an MTA (mail transfer agent - a must have even if you set up Moodle to use SMTP correctly):
ps aux |grep sendmail
ps aux |grep postfix
ps aux |grep exim
To test the sending of mail from server, one could install something like alpine - text based command line mailer. Config it and send a test message to yourself.
Have you checked server logs? (the following for CentOS) found normally in /var/log:
or apache logs ... normally found in /var/log/httpd/
'spirit of sharing', Ken