Interfacing without an Interface
If the switch is connected to a pin of a microcontroller, then the lead will have very little current, and will not be able to drive a relay or otherwise be current-sensed.  However you should be able to sense whether the switch is pressed or not by looking at the voltage of the switch terminals.

With the switch open, measure the voltage on the two pins.  One might be ground (0), and the other a voltage like 3.3v or 5v, usually referred to as Vcc  Then close the switch.  Depending on whether there is a pull-up or pull-down resistor, one of the leads will change to match the other (both will now be ground, or both will be Vcc).  You can then use that lead as the indication of your switch closure.

As you proposed, you can use a solid-state relay across the contacts to simulate the switch closing.