I am wondering, how does one actually conduct position control on a voice coil actuator? In other words, to hold the shaft static at position X, does it require a constant voltage/current?
A constant voltage or current will hold the shaft at a constant position provided the load, angle, and temperature don't change. To control in that fashion is generally called "open loop" and can work well enough for some applications.
For precise position control some form of position feedback is required to adjust the output so that the position error is reduced. There is no feedback built into the units in your link.
Or, can you remove the input voltage/current, and have the shaft not move?
There is no mention of a shaft brake on the datasheet either. The brake, if supplied, would require additional wiring as you couldn't expect the brake to release on low voltage operation (for a small movement).
Response to comment:
If a brake were to be included, what would be the brake's technical name?
I'd be looking for "holding brake" or similar.
Would it be a linear magnetic brake?
No. Linear magnetic brakes apply a decelerating force proportional to the velocity of a conductive plate through a slot in a magnet. At zero velocity the have no holding force. You'll need a mechanical holding brake.
Also, to interpret what you're saying in the 2nd half of your post... If the brake were added, it would prevent motion at low voltages because the power input to the VCA would have to be enough to generate a magnetic field that could overcome the magnetic field from a brake
Let's say we had a 12 V actuator with a 12 V brake. The brake requires some voltage - say 8 V - to release it fully. If the brake is sharing the actuator wires then the actuator won't move well or at all until the voltage reaches 8 V.