What you can do is

1) hook up Web Browser events, using for example this sample How to sink HTML document events for WebBrowser host

2) once you get a hold on the loaded IHTMLDocument2 document interface, just read the value of the cookie property:

CComPtr<IHTMLDocument2> pDoc // get document from event as shown in the sample.

// read the cookie
CComBSTR cookie;
hr = pDoc->get_cookie(&cookie);

More on this here: Handling HTML Element Events

Note: for NSAPI I suggest you write another question with other tags as this is a totally different world.