Roll-to-Roll (R2R) printing techniques are promising for high-volume continuous production of substrate-based products, as opposed to sheet-to-sheet (S2S) approach suited for low-volume work. However, meeting the tight alignment tolerance requirements of additive multi-layer printed electronics specified by device resolution that is usually at micrometer scale has become a major challenge in R2R flexible electronics printing, preventing the fabrication technology from being transferred from conventional S2S to high-speed R2R production. Print registration in a R2R process is to align successive print patterns on the flexible substrate and to ensure quality printed devices through effective control of various process variables. Conventional model-based control methods require an accurate web-handling dynamic model and real-time tension measurements to ensure control laws can be faithfully derived. For complex multistage R2R systems, physics-based state-space models are difficult to derive, and real-time tension measurements are not always acquirable. In this paper, we present a novel data-driven model predictive control (DD-MPC) method to minimize the multistage register errors effectively. We show that the DD-MPC can handle multi-input and multi-output systems and obtain the plant model from sensor data via an Eigensystem Realization Algorithm (ERA) and Observer Kalman filter identification (OKID) system identification method. In addition, the proposed control scheme works for systems with partially measurable system states.