Is it possible to have an automatically updating link to a user's profile picture? Such as by giving each user one "slot" for an avatar?

That’s basically the about page isn’t it? Like About - Discourse Meta

The correct way of keeping user info synchronized between different system is using DiscourseConnect - Official Single-Sign-On for Discourse (sso). However that may be too much if all you want is a static page with the current forum admins. In that case you can maybe have a daily/weekly script that parses /about.json and updates it?