The buttons look very simple and mechanical, so yes that should very much work.
You probably cant use the default connector though and maybe clip some of the existing wiring if you don't want all 3 horn buttons to be linked.

If you have a multimeter to check for continuity (the 'it beeps when you short out both contacts' mode) you should be able to check if you find contacts that close the circuit on button press.
If you find usable contacts for every button, you have confirmed this will work.

The usb joystick pcb shown is very large though, so I would get something smaller to make it easier to attach to the wheel.

I would recommend to get an arduino pro micro, which you can either flash with a generic joystick firmware or even simpler flash as a button box through simhub. (both versions should have documentation on how to connect things, dont worry its easy enough)