Well,
then I recommend to change your hosting provider!
Whatever is meant by 'semi-dedicated host' ... it's of no use if it doesn't even come with SSH access - because you'll need that not only for exactly examining things as asked by Arsalan but for database backups, too.

I do not have access to the console on the server, as i mentioned before the site is hosted,

The Cache is enabled
and the OSSN version is 5.5

The images you provided isn't enough. Before you mentioned that it only happens when you activates chat now it happens with every activity. So we are still not sure what exactly is wrong.

Try to run following command on your server

ps aux --sort=-%mem | head

Which should show correct processes using most memory.
However you have not provided other details like

1. OSSN version you are using.
2. OSSN cache enabled or not?

Sorry for the delay,
the only information i have access to,is similar to hte above images,

It spikes every time some user performs any kind of action, whether is a post, a redirect.

How can it take so long to provide the requested information, Iason?

What is crashing exactly?
Please provide the logs reflecting these crashes (e.g. syslog, logs of webserver, database server)