Recent advances in Model Predictive Control (MPC) algorithms and methodologies, combined with the surge of computational power of available embedded platforms, allows the use of real-time optimization-based control of fast mechatronic systems. This paper presents an implementation of an optimal guidance, navigation and control (GNC) system for the motion control of a small-scale electric prototype of a thrust-vectored rocket. The aim of this prototype is to provide an inexpensive platform to explore GNC algorithms for automatic landing of sounding rockets. The guidance and trajectory tracking are formulated as continuous-time optimal control problems and are solved in real-time on embedded hardware using the PolyMPC library. An Extended Kalman Filter (EKF) is designed to estimate external disturbances and actuators offsets. Finally, indoor and outdoor flight experiments are performed to validate the architecture.