I am not trained at all at doing admin for phone systems but I am very good at researching and testing things.
I am an office assistant at a medium-sized non-profit and we have the Inter-Tel 8520 Phones and 1 admin 8560 phone.

This system was donated to us several years back and there is just 1 company left in AZ that even works on these systems that I could find.
We have about 8 stations set up now and I found about 20 unused 8520's in a closet and suggested that we put an extension in all the rooms we have for emergency/security reasons. I am totally comfortable figuring out how to wire the phone from the backend equipment but can I add this from the admin station or do I need to hook up a laptop to the backend stuff and have special software?
Thank you!

There are plenty of Mitel Dealers in your area to work on this. Give one of them a call.

In fact Inter-Tel used to reside in AZ. You need session manager for admin purposes.