I don't know about Stanford in particular, but my experience so far has been that if you just volunteer to do stuff (and maybe have rear cover from one higher-up, but that's often not really required), and don't need any budget, people just let you do stuff.

At work, I started a study club, and started teaching a programming course, all on company time. No pre-approval. Nobody objected.

And the universities I've been to tend to be even more liberal than companies in such matters.

So an undergrad teaching a course doesn't seem questionable to me.