This depends on what/how the extension is accessing the network. They have a background page and they can inject scripts on visited pages. If they are injecting scripts which then make the requests then it would show up in the DevTools on the page you are visiting, but this is uncommon.

Instead it is far more likely for the extension to make the requests from their background page (which has elevated permissions) which is essentially its own page with its own inspector. You can inspect each extension individually by going to your extension listing, enabling developer mode and inspecting the background page of the extension you suspect.

Right, but I think the parent was suggesting a way to see historical URL accesses by extensions so you could audit them.

You can see EFF PrivacyBadger code getting injected in the source tab in dev tools, not positive about network requests. It wouldn't be hard to test though, just make a simple extension that does XHR to example.com and load it up locally. Chrome extensions are surprisingly easy to write, and there are some simple tutorials if you google around a bit :)

EDIT: here's how to inspect extensions you're curious about: https://developer.chrome.com/extensions/tut_debugging