If I understand you correctly, you are approaching this as a Regression Analysis. Namely, condition Y as a function of X (or multiple X's).

So, you are producing widgets (Y). You are wanting to identify the different variables (X) that will affect (Y) during production. However, you cannot manipulate any of the X values while it is running. Is that correct?

First, approach it from a common sense perspective. If you are making widget, on the surface what factors seem reasonable that would affect it? This is important. You cannot have a bunch of variables that appear with numbers to be related, but have no association. Your variables need to make sense.

Begin measuring these variables, and the states of your Y (output). With even data, you can develop a regression equation (or whatever type of analysis you would like).

There is more than one way to do research. Experimental design is where you manipulate one variable to observe the effect on another. However, a lot of research does not allow variables to be manipulated. Therefore, a theoretical case is made for a relationship, measurements are made, and the case is supported or not supported.