Smarter Document Management links between three WordPress sites sought

There is nothing in WP that specifically deals with cross–site communication. Every WP site is an island by design.

However there are plenty helpful bits and pieces to rig suitable solutions for many use cases.

As far as I follow your case I would imagine something like this:

  1. Central sites hosts documents (in whatever way is convenient for you).
  2. Central site publishes machine–readable (JSON for example) registry of documents with their links. Implementation could be as special template or whatever, many ways to do it.
  3. School sites pull registry (HTTP API), cache it (Transients API), and display.

Building blocks are (relatively) basic, but this will ensure that central site is in control and linking sites are up to date and don’t rely on humans to keep up with the documents and links.