You could clearly force a browser to come up from a WinApp, but I certainly don't see how that helps you IN THE LEAST to get any value from the Web app. They will be running in separate process spaces (I don't think you can ask the win app to bring up a browser in the same process....but I could be wrong about that) and so won't have access to each other's memory spaces.
A *much* better way to do this is to have the web app provide a web service that you can invoke from the win app. That is, design ahead of time for this situation.
Failing that, you could have your win app *pretend* that it is a browser (that is, it could make an HTTP request of the server, passing the appropriate credentials and request information). How hard or easy that is depends on how the web app's security is handled. Having done this, you'd still have to "screen scrape" the HTML returned from the web app looking for the relevant data.