Typically, the desktop application would act as a client here, and contact the server (possibly the same server that runs the web app, possibly a slightly different server intended just for API access). It is possible for a browser application to talk to a desktop application, by having the desktop application install itself as a custom protocol handler - but this may require more permission than you are allowed on the desktop machine, and it is relatively complex. Having the desktop app talk to a server, and the browser app talk to a server, but not directly to each-other: is a simpler setup.

If you mean to do this without any kind of browser session on the same machine, then:

• either the desktop app needs to be running as a client, connecting to a server, and listening for messages, or:
• the app needs to be a server, and open an inbound TCP/UDP port, to listen for connections/messages

The second option presents much more security issues; in a lot of corporate scenarios, this could be disallowed and blocked by corporate firewalls, or could even have the network security folks arrive at your desk with a cardboard box for your things.

1. You will need to setup your server somewhere in the internet or inside company LAN.
2. Desktop app should connect to this server and keep connected, waiting for a data. Or you can check for a new data by a timer, if your scenario allows some delays.
3. When you need to make a request, you send data to your server (may be including client id or something similar to identify the exact instance of the desktop app).
4. The server forwards data to the desktop app and then forwards the app's response back to you.

Note: you will not able to connect directly to the desktop client without intermediate server, as most likely the company computer does not have "white" ip and is located behind the company's NAT.