How Do I Programmatically Get Caller ID on Current Call with Five9?

Question

The docs appear clear as mud on this. I'm building a custom CRM and need a way to get the phone number (caller ID) of the current incoming call. The workstation has the Five9 application installed and the Desktop Toolkit. Now what do I do in the web browser to get that information? My idea was that I'd have a search form and one could click a button called "Search by Caller ID". The web browser would then connect somehow to Five9, get the current phone number of the current call, and send it back to the web server to show records that matched that phone number.