Bind the checkbox click event?

If I understand correctly the webhooks allow me to do something on the server side when triggered, not within the UI

You’re correct.

Are you building a Power-Up, browser extension, or something else? There is no way to do what you’re looking to do via the Power-Ups platform. I assume there is some way to do it via a browser extension, but the DOM isn’t guaranteed to stay the same forever: Trello and Custom Chrome Extension - #2 by bentley