How to contol the height of blade in a vinyl cutter machine?
Most Vinyl cutting applications do use a simple moving armature to raise and lower the cutting tip.
This means the cutting tip is pushed down with a fairly constant force when cutting, it's position (height) is NOT controlled. Using this method it can handle variations in the material thickness without problems.

If indeed the vinyl cutter does use the A3955, then you would have to either excise the signals from the controller board or build a new controller for the Z axis control.
The A3955 is used to provide accurate current control and the ability to pull/push a magnetic armature.
I assume for the X,Y axis you intend to build an Arduino controller for these and not break into the cutter control board. The same strategy for the Z axis would seem best.

The biggest problem you have is understanding the Z axis current profile they used. It may be quite complex and you'd need an oscilloscope and current probe (or series resistor) to really figure it out.