Haagimus

those commands were all run from the root site directory. Should I be doing that from a different directory?

@Jim M here are the modules you asked about.

sudo httpd -M | grep rewrite_module
  rewrite_module (shared)
sudo httpd -M | grep setenvif_module
  setenvif_module (shared)

My server admin is walking into a meeting right now, I will get that information and update this post when it is available.

Apologies, how do I check that?

Title says it all. I have completed the required steps to enable the API services with no success.

The file ".htaccess" has been uploaded to the /api folder in my sites directory (/var/www/html/api). When I attempt to access this endpoint however I get the below error when clicking continue after. 

The API endpoint is not giving the expected response. Check you followed the instructions correctly. 
The URL being tested is: https://10.253.0.105/api/index.php?/core/hello

Can anyone help me figure out what is going on and why I cannot access this?

There are no log files or entries for the errored attempts to connect.
The system is running inside of a company VPN network.
The system has no custom folder structuring in the OS.