If you control the websites that you want this functionality on, you could achieve this by setting each application to install a ServiceWorker. In a nutshell, service workers run as a proxy server within your browser. They can do things like notify you of updates even if you don't have the website open.
Within a ServiceWorker you are able to set up event listeners that can do some asynchronous task on behalf of the client app. This includes the fetch
event which is fired every time the web page makes a request.
Here's a write up on someone implementing a ServiceWorker who also needed to intercept network requests. You could follow most of this and just alter the logic when inspecting the request type. At that point you could add any special headers before dispatching on the applications behalf.