In order to mitigate economical, ecological, and societal challenges in electric scooter (e-scooter) sharing systems, we develop an autonomous e-scooter prototype. Our vision is to design a fully autonomous prototype that can find its way to the next parking spot, high-demand area, or charging station. In this work, we propose a path-following model predictive control solution to enable localization and navigation in an urban environment with a provided path to follow. We design a closed-loop architecture that solves the localization and path following problem while allowing the e-scooter to maintain its balance with a previously developed reaction wheel mechanism. Our model predictive control approach facilitates state and input constraints, e.g., adhering to the path width, while remaining executable on a Raspberry Pi 5. We demonstrate the efficacy of our approach in a real-world experiment on our prototype.