Emerging connected vehicular services, such as intelligent driving and high-definition (HD) map, are gaining increasing interest with the fast development of multi-access edge computing (MEC). For most time-sensitive and computation-intensive vehicular services, the data offloading process significantly influences the capacity and performance of MEC, especially when the number of connected vehicles is enormous. In this work, we consider data offloading optimization for a large-scale automotive MEC network. The problem is challenging due to the large number of connected vehicles and the complicated interaction between vehicles and edge servers. To tackle the scalability problem, we reformulate the original offloading optimization problem into a Mean-Field-Game (MFG) problem by abstracting the interaction among the connected vehicles as a distribution over their state spaces of task sizes, known as the mean-field term. To solve the problem efficiently, we propose a G-prox Primal-Dual-Hybrid-Gradient (PDHG) algorithm that transforms the MFG problem into a saddle-point problem. Based on our developed MFG model and G-prox PDHG algorithm, we propose the first data offloading scheme whose computation time is independent of the number of connected vehicles in automotive MEC systems. Extensive evaluation results corroborate the superior performance of our proposed scheme compared with the state-of-the-art methods.