Pick and place machine

Stepper motors require a differential offset square wave input. This is commutated by the driver every time it receives a step pulse, in the direction given by the direction input. For something relatively low precision like this you'll probably use full stepping, maybe 1/4 stepping for smoothness. Fractional stepping is where the square wave trains are chopped up even further with PWM into a kind of square sine wave, and the teeth on the rotor don't quite fully line up with either full-step pole on the stator. Makes it easier to smooth out acceleration.

Assuming your PLC doesn't output step/dir signals, and I don't think it makes sense to even attempt to make it unless it's specifically designed to do so, you'd need an intermediary between the PLC and the driver to convert "forward" and "return" signals into usable, ideally acceleration-planned (to stop jerk from dumping your parts), step/dir trains. This would probably be a pair of Arduinos, one running grbl, one taking in PLC signals and outputting pre-written serial commands to the grbl unit. It's a little bit of writing basic C, if you're up for that.

Personally I'd probably just use spring-buffered, maybe even spring-return, pneumatics, maybe with a belt-based telescoping pulley mechanism, if it's such a short, permanently fixed distance and you're already familiar.

Discussed a similar kind of project (though required sinusoidal motion rather than point-to-point) here, which might be relevant:

 

Yeah I can't gauge space constraints without images, so that's your call, but the latter could be pretty easily solved with a second, much smaller cylinder operating a stop block or sear that ensures a dead stop in one or both directions of travel.

That actually looks like a pretty interesting unit, especially since it can take in quadrature encoding- presumably both rotary and linear, shouldn't matter- for extreme positioning precision if you were to add a scale to the axis. It'll certainly solve any problems with acceleration planning and jerk avoidance, but you may have to change from a passive optical sensing system to more of a programmed distance system to avoid the need for sudden stops. Even without encoders, well-calibrated steppers running a backlash-free belt drive system should have no problems hitting +/-0.1mm all day long.

I'd generally be inclined to agree, unless you can find a very high speed ballscrew inexpensively- something in the 30-50mm pitch range, but not too large a diameter to reduce inertia. Otherwise, a beefy HTD belt or similar will run an incredibly long time with near-zero maintenance.

I'd go servo over stepper too, if you can. The ClearPath SK series will accept step/dir signalling whilst also providing the speed/torque/inertia/noise benefits of servo drive. Size-wise, anything that's about NEMA 23 or 24 equivalent (57-60mm face) should be just fine for a few pounds without overheating from all the stop/starts.

Wheels are fine for hobby gantry routers and some other low-speed projects, but for long-term reliability and rigidity in automation, I'd use linear rail and shafting all day long (literally!).

Motors have torque curves available. Torque = force * distance. Distance is the pitch radius of your drive pulley. Force gives you your acceleration, speed and distances via the equations of motion, etc etc. Just a case of number-crunching to find a motor that can output the torque you need, up to the speed you need, at the voltage you're using (probably 24V, maybe 36V or 48V if you're lucky; not all of the smaller motors accept voltages that high, but they'll give you higher accelerations without skipping steps). At a certain point you can drive yourself nuts though, sometimes you just have to pick something to try.