I see that I can run an NIS from my home computer to help out the network but I’m wondering if it’s possible to host it on my web host?
While you can host a NIS from your home, the ideal hosting method for NIS would be a VPS (virtual private server).
So yes, you can certainly run a NIS on a server 
This is a really solid guide, just ignore the super node parts, a super node is just a NIS with monitoring software added. NEM Supernode Command Line Tutorial for Debian 8.4