My boys attended elementary at a private Christian school. It's owned and managed by our church. Like most businesses, it wasn't paperless, and not even close to being able to work remotely. Yes, a church and a school can operate it's business administration remotely. It absolutely can. But why would it need to?
The one thing all businesses have in common is operations. Every business operates in some fashion. The key is being able to operate under any circumstance. Just ask my church how difficult it was when their admin building burnt to the ground. Cash, files (years of files!), computers, all burnt.
They weren't prepared for a disaster. Are you? I know I am.