Computer controlled rotary switch
You can get a reed relay, such as this one, with contacts rated at 200 VDC @ 500 mA, and a nominal coil voltage of 5v.  The datasheet is here.

The advantage of a relay over a transistor solution is it would provide total isolation.
Also the easiest way of integrating into your current system, just wire the relay contacts across your switch contacts.

Since you are switching a fairly low current, you might want to use a relay with two Form A contacts and wire them in parallel.  This relay is identical to the one above except it is DPST (and costs about twice as much).