How to switch something else on with an arduino
To press a switch, you'll need some kind of expensive solenoid or servo, and mechanical linkages, and mounting hardware, and trigger pressure characterizations, and power-loss state characterization, and a 12V power supply, and driver circuitry, and it all adds up to a lot of work to push the physical trigger.

If I were you, though, and had access to the hardware, I'd characterize the circuit the trigger switch is attached to (is it normally open or normally closed is really all you need to know), and attach a simple relay or MOSFET to an I/O pin on a microcontroller (I'd suggest the relay, because it's closer to the original circuit, avoids any voltage referencing issues, and has a mechanical spring to ensure that if your circuit goes dead, the trigger won't go off).  Then, remove the switch from the gun, and attach your circuit.