In unmanned weighbridge Edge services, determining whether vehicles are correctly positioned on weighbridges from video streams efficiently at the Edge is a critical challenge. Traditional monocular deep models for vehicle localization are not lightweight and accurate enough for deploying on Edge devices. Meanwhile, existing inference scheduling methods often neglect multi-stream concurrency and effective collaboration among Edge devices. Therefore, we propose a monocular indirect vehicle localization method based on a lightweight keypoint detection model, optimized for NPUs through customized neural operators, together with a Multi-stream video Distributed Inference and Scheduling (MDIS) framework that integrates frame stitching, adaptive sampling, and weight-based load balancing. The proposals are evaluated on a real Edge device-based cluster, and experimental results show that our approach reduces parameters by 33.46%, computational cost (FLOPs) by 38.63%, and improves the inference speed on Edge devices by approximately 33% without decreasing detection accuracy. MDIS is able to obtain the lowest latency and fail rates. The codes are available at https://github.com/JiaQiChen-Lucas/vehicle_localization .