Controlling cardreader numpad input with microcontroller
Those inputs are probably matrixed, since I don't see any obvious ground traces.  Matrixing means that you drive one side of some switches low, and read the other side to see which one is pressed, then drive some others low and read them with the same input lines.  Matrixing allows this particular keypad to be read with 9 I/Os instead of 20, but it means driving high and low in an uncoordinated manner can result in errors.

You could figure out the matrixing sequence--which side of each button is driven vs. which side is read, and use AND or OR gates to handle the multiplexing--but it would probably be much easier to use analog switches driven by your controller.  Those would simulate the operation of the mechanical switches.

The analog switches can be connected across the top and bottom (redundant) terminals, or if your soldering skills are very precise, connecting across the fingers is also an option.