Whilst you are faced with a multitude of issues, don't get lost in the weeds by details when you are trying to untangle the past to move it forward.
A simple spreadsheet to track hardware, licenses and other details like location, specs and primary contact is a perfectly reasonable starting point.
I say that because you don't know what you don't know yet. You might for example discover that some shops are doing their own thing, regardless of company policy.
Creating a ticketing system is useful to track stuff for everyone. I settled on trax with web access to people who need it, but the computer literacy levels might prevent some from using this.
Burnout is a very distinct possibility in an environment like this, so make sure that you set aside time for you to think. Call it a meeting, call it an on-site visit, whatever you do, take time to think.
Also, remember to backup your work. It's not unheard of for it to vanish unexpectedly if you are perceived as a threat.
Source, I've been working in this profession for 40 years.
I’m not in tech anymore but must comment that I work at a major company in a dynamic field with young, ultra-qualified, ultra-smart personnel that is not horribly computer-unsavvy but I still think I’m the only one in the whole company who opens IT support tickets via a system instead of caling, even though IT pushes the system and even though you get good support via the system and horrible people by calling
The ticket system is for the IT department, allowing it to track activities, keep abreast of open tickets, build a knowledge base and share information with colleagues.
Users benefit from this indirectly.
Of course, some managers use ticket systems to manage performance metrics. That doesn't work, but they'll never learn.