Is there such a thing as a switch that can be actuated automatically?

What you are looking for is a rare beast. Honeywell produce a toggle switch (2 position and 3 position) that can be remotely reset by removing the holding current - this releases the magnetic force exerted by a small solenoid and the switch returns to off or centre. This may do what you want but, I suspect that you would want to be able to toggle it at will remotely: -

Why can't you easily find one that can toggle in both directions? Complexity and performance expectations leading to unfeasible cost and therefore NO foreseeable market is the main reason. Hey it was hard enough to find this one let alone one that can be operated at will in both directions remotely.

You can get a slide potentiometer that is motorized. It has a 100mm stroke, which you'll want to restrict to maybe 15-30mm, but with a suitable knob, and a microcontroller you can emulate your motorized switch.

You can even provide feedback and make it feel like a snap action switch. When it's moved away from one position, it pushes back a little bit, until it's moved past center, then it accelerates to the other position and stops. You can define your own spring constant for the snap action, up to the force of the motor control, anyway.

It would take up a lot of space, but might fit your needs.

The closest thing that I can think of are Syndyne SAMs (Stop Action Magnets). They are designed and primarily used for organ consoles, where the stops need to be set manually by toggling the SAM, but where their states may also be saved and recalled at the push of a piston (small button under the keyboard).

I'm not going to repeat the specifications because all of that can be found on the website I linked to, but basically each has an electromagnet to pull the switch on when powered, and another electromagnet to pull the switch off. They operate very quietly and smoothly, but still have a gentle tactile feel.

Syndyne also makes Solenoid Draw Knobs, which accomplish the same task in a slightly different fashion.

Last I checked, the SAMS cost somewhere around $30 to $50 each, though I'm not sure if Syndyne sells to individuals. They can sometimes be found on eBay, and some organ salvage businesses might have some to sell as well. I'm not sure about the SDKs.

Also, I realize this post is old, but perhaps someone might still find this info useful.