You should be fine. Yuzu checks the Github repo for updates which is now down. If you're still worried you could download it via Flathub and disable network access via Flatseal or terminal.

I have the opposite question. How do I make sure that Nintendo knows every time I run yuzu?

boom!

Lots of people have already forked it. I think it's even backed up on the Internet Archive too. You could also check the Yuzu subreddit as well.