Leverage the REST API for extension/WordPress communication.
EDIT
As discussed in the comments, creating custom AJAX handlers is probably a more straightforward and “more simple” approach.
In order to implement your project using the REST API, you would likely have to implement a custom post type as well as a custom REST endpoint controller class. Using a simple AJAX approach, however, you only need implement a AJAX controller.
As an alternative to implementing a heavier browser extension, you might also consider implementing a bookarklet instead – that is, a chunk of JavaScript code which is executed through a browser’s address bar, and thus can be saved in virtually any web-browser as a “bookmark”. This is how the “Press This” feature shipped with WordPress functions.
In either scenario, it’s possible that you’ll need to configure the endpoint(s) which communicates with the extenstion/bookmarklet to allow HTTP requests originating from virtually anywhere. An HTTP request to a server originating from another domain than the server itself is commonly called a “Cross-Origin-Request”, which is managed through web-server and application rules regarded as “Cross-Origin Resource Sharing” or “CORS“