A fractional factorial is one of many experimental designs that could be used here. Since you don't specify whether or not you want to look for interactions (and what type of interactions), since you don't specify whether or not you want to look for curvature, and since you don't specify how many runs you can make of this four step process, and you don't specify other things that may or may not be important in an experiment, then all I can say is that you could use a fractional factorial design, but other designs might work better.
> Also, is there a mechanism that could allow me to correlate analytical data from
> output/input of one step with the analytical data from the product?
Let me make sure I understand ... we are no longer talking about a designed experiment, but simply analysis of data collected from this process, right?
In that case, correlation analysis should be able to do this.
> We have a debate going with some of our group arguing that the tolerance
> intervals of a given step output can serve as a predictor of the product quality.
Tolerance intervals are basically a function of the variance at a particular step. Therefore, you could use the variance rather than the tolerance intervals as a predictor. I have never seen tolerance intervals used as predictors.
> I argue that the remaining process steps aren't accounted for in those tolerance
> intervals and thus limit their ability to predict the final product quality.
Any single predictor from Steps A, B, C and D necessarily doesn't use information from later or earlier steps, and so their ability to predict may be limited. Or its ability to predict may not be limited. It depends on your process. But, if you know you have a multi-step process, you will be better off building a model that has all of the predictor variables in the model (plus interactions, curvature as appropriate) from all of the steps. You wouldn't normally model on a single predictor variable's effect on final product quality.