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Rapid transit of emergency vehicles is critical for saving lives and reducing property loss but often relies on surrounding ordinary vehicles to cooperatively adjust their driving behaviors. It is important to ensure rapid transit of emergency vehicles while minimizing the impact on ordinary vehicles. Centralized mathematical solver and reinforcement learning are the state-of-the-art methods. The former obtains optimal solutions but is only practical for small-scale scenarios. The latter implicitly learns through extensive centralized training but the trained model exhibits limited scalability to different traffic conditions. Hence, existing methods suffer from two fundamental limitations: high computational cost and lack of scalability. To overcome above limitations, this work proposes a scalable distributed vehicle control method, where vehicles adjust their driving behaviors in a distributed manner online using only local instead of global information. We proved that the proposed distributed method using only local information is approximately equivalent to the one using global information, which enables vehicles to evaluate their candidate states and make approximately optimal decisions in real time without pre-training and with natural adaptability to varying traffic conditions. Then, a distributed conflict resolution mechanism is further proposed to guarantee vehicles' safety by avoiding their decision conflicts, which eliminates the single-point-of-failure risk of centralized methods and provides deterministic safety guarantees that learned methods cannot offer. Compared with existing methods, simulation experiments based on real-world traffic datasets demonstrate that the proposed method achieves faster decision-making, less impact on ordinary vehicles, and maintains much stronger scalability across different traffic densities and road configurations.