Interfacing with (Two) Full-Bridge PWM Motor Drivers to Control a Stepper Motor
The A3959 looks like it's intended for controlling a brushed DC motor, where applying voltage in one direction produces continuous rotation, so the PHASE input does act as a forward/reverse control.
  For a stepper motor, you need to provide the two phase signals in the right sequence to get movement in either direction  
 (this is figure 3 from your link - https://www.embedded.com/print/4217719 

As a result you've got two of these driver chips, and need to produce this signal to both of them. The L297, with a clock and direction input, will do that. You'll only need to use two of the outputs, such as A and C in 'normal mode' since the A3959 is generating the complementary drive to the other side of the bridge.

"Is this similar to a clock signal? (Note: ITRIP is already calculated perfectly for the motor current when VREF is a full 5 volts.)"
  No, it is a DC voltage that sets the level at which the current limit operates.  

"If I were to supply 100% VREF, would it build up to 100%, produce one single step, trip and then wait the fixed time off before going again?" Yes, see the sawtooth in the figure? That's the current rising to the limit value, and being chopped, when it falls for the fixed time period, before the bridge transistors  switch on again. The current then will keep bumping up to the limit value.

"Would a constant VREF keep the motor going, pulsing automatically?" No, the motor only rotates one step with each phase actuation and will stay there, actually locked into that position if the phase current stays on, until the next step is switched. There are half step strategies (detailed in the L297 datasheet) where one phase has no current on alternate steps, but with the controls available on the A3959s you don't have that option, you have to do full step - unless you can fudge a disabled state by dropping the Vref input low. It's not meant to operate that way, there's a separate ENABLE pin intended for PWM control.
  At this point I'd be tempted to program a PIC to generate the pulse sequence, but the L297 will get you a working drive.