Moodle 3.1.x is long term support ... meaning it can get security fixes/updates only. It would be advised, however, to update the site to the highest/most secure of 3.1 which is now 3.1.7+. IHHO, the best way to do that is via git. That, however, does require ssh access to the server.
CentOS 6's end of life is 2020.
And there is another piece to moodle you've not mentioned ... that of the DB server/versions. On a typical CentOS 6 am guessing that would be MySQL.
Error 500's are a little troublesome to resolve as that error is a generic catch all but one will get clues by looking at apaches error log which on typical CentOS 6 is in /var/log/httpd/error_log
Many times error 500's do relate to ownerships/permissions on moodle code and/or moodledata directories. On a typical CentOS 6 server the web service runs under the user 'httpd' ... group 'httpd'. CentOS 6 typically installed and runs seLinux (security enhanced linux) - which if not configured correctly could give Moodle some grief.
Am aware that enties such as the one where you work don't usually have lot's of cash for IT support and you are probably wearing many hats ... of which Moodle is only one. The real bottom line is how familiar are you with Linux (and the OS you are running)? Afraid that the most efficient way to admin your server is via ssh and command line.
Like any Linux, probably best to consult the source of the distro, so the following link might be a place to start educating yourself on CentOS 6:
https://www.centos.org/forums/viewforum.php?f=3
Also, one could turn on debugging in Moodle, then attempt to do what produces an error, then copy and paste the output of debugging back here in this thread for others to see ... even suggest as to what to check etc..
Will say this ... makes no difference how large the site nor what Linux OS Moodle runs under, it's best to keep the OS up to date as well as Moodle up to date within a series that's still supported.
'spirit of sharing', Ken