Model-free (MF) strategies have emerged as a promising solution to challenges associated with modeling errors and uncertainties in predictive control (PC) systems. In this context, MF-PC schemes utilizing Auto-Regressive with eXogenous input (ARX) time-series models offer a flexible approach for online predictor construction using input-output data. However, the design of the polynomial orders within the ARX structure is critical, as it determines the balance between model accuracy and computational cost. This article presents a novel systematic method for designing ARX polynomial orders in MF-PC, based on well-established statistical criteria. Unlike traditional trial-and-error approaches, the proposed method offers simplicity and efficiency, allowing for accurate designs using general information about the controlled system. To demonstrate its feasibility, the proposed method is applied to the design of a MF-PC voltage control of a grid-forming inverter (GFI). Experimental trials conducted in a laboratory-scale GFI validate the effectiveness of the proposed method, delivering robust and accurate reference tracking under set-point and load disturbances. A comparison with conventional model-based PC highlights the advantages of MF-PC with an ARX predictor designed using the proposed methodology in the presence of model uncertainty.