I don't think any browser vendor would expose an API to allow this.

I think the best you could manage is a system administrative profile that forces the browser to install the extension

Or, depending on how serious you are, you could modify the browser source to make it unremovable, at least for Chromium or Firefox

You could write your own OS software that polls your browser for installed extensions and then nukes your internet connection through the host file if you disable the extension. Cold Turkey kinda does this by automatically closing your browser during a blocked session if you disable their extension.

The challenge is that there's always a workaround. The added friction might be enough to fulfil your need though.