I have a desktop application and I am using the Process.Start property in this application. Can I somehow send a request to this program over the web?
Note: I cannot install iis or similar things on the device as it is a company computer.
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:
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.
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.