Well, in this case the modifier is the "Array" and I would like the measurement of the "Constant Offset" of the Y axis to correspond to the Y axis of the "Transform Orientation".
I don't know if there's an easier way to do what I want, but what I thought led me to this idea, but I don't know if it's possible.
To answer the question you asked, no. The transform orientation is entirely contextual (currently it is set to “Face”), while modifiers must continue to work no matter what the context is. They are applied on top of all other operations procedurally. You can use the object’s local axes, or the scene’s global axes, but not the currently selected face’s axes as that will change the moment you select a new face.
I suggest you ask the broader question of how best to achieve a certain modeling structure. You seem to have made a few wrong assumptions and hit a dead end when there was a path forward further back.
