I would go for separate input devices.
1) Gamepad for Xbox 360 is USB compatible with Windows PC's, but might look a bit "gamer/unserious" depending on your product and customers.
2) Touch screen, perhaps just a very small one, might be a choice?
3) External numpad can also be attached. That gives you at least 10 buttons.
That would be my first 3 choices.
You could go for option #1 and the take them apart to get to the buttons inside. There are many buttons on an Xbox 360 gamepad.
at the front you have 2 buttons left and 2 buttons right.
On the top you have 4 coloured buttons for actions.
For your left thumb you also have a digital direction-pad with 4 directions + combined ofcause for diagonals.
That gives you at least 3x4 buttons from hardware connection. Question is if you want to mess around with your own buttons and cords, because you ought to be able to extend the cords of these onto some kinda arcade board of your own.
The controller itself is fully recognized by Windows, once plugged in.