It can be done using a programming language like Visual Basic that can handle calling GUI events on other applications. In other words, you could write a Visual Basic program that will find the system ID of the Icom Control window, locate the correct button(s) to "click"/"push" by calling an event, and it would appear to the Icom program that someone literally clicked the button on their program.
I've only fiddled around with programs that do this in the past, but it's definitely possible. You'd have to know your way around Visual Basic and use a program like Spy++ (included with VB) to find out the name of the form (window), find the name of the button (in the other program), and raise the Click event.
The only other hurdle is to write the code to enable the client and server programs to communicate over the Internet, which isn't that hard.
Alternatively, if you know the commands to control the receiver directly via serial/USB, you can just ignore the button clicking aspect and send the serial data right to the receiver. Your only hurdle is to add the Internet connectivity to it, so you can send commands from a remote computer and have the local computer pass the serial right to the receiver. Much, much easier.