As said by someone else, at last something in my profession. The idea is to document a process without implying that a person must explain their job and thus risk losing it. How to get to the solution is to advertise that you want to incrementally improve a process and hold a kaizen event. (Wikipedia it.) The first step is to establish an "is" flow chart by function (individual person) with either the "modern" sticky notes or 3x5 cards (yes, I am THAT old). Secondly, ask for "should be" changes from the stakeholders (department managers and the people who do the work). Third, try to achieve consensus (the real trick) to any changes and document the new flow. This document should be controlled by the Training/HR Manager with either an online process or written SOP. (I work in the drug field and the FDA can get sticky.) The stake holders (managers) are now required to have the people that report to them issue "vacation" support, i.e. how to cover that process only. After all, it's the managers responsibility to make his department work at all times. Function by function, process by process, things get documented. When you're done you can start over again and try to incrementally improve the processes again, or document the new ones.
I've spent 30 years in what is now called supply chain management. The prime point is to distribute responsibility to the correct department and make the manager responsible for his department's actions and results. That's why they get the big bucks and headaches.