In vehicle trajectory tracking tasks, the simplest approach is the Pure Pursuit (PP) Control. However, this single-point preview tracking strategy fails to consider vehicle model constraints, compromising driving safety. Model Predictive Control (MPC) as a widely adopted control method, optimizes control actions by incorporating mechanistic models and physical constraints. While its control performance critically depends on the accuracy of vehicle modeling. Traditional vehicle modeling approaches face inherent trade-offs between capturing nonlinear dynamics and maintaining computational efficiency, often resulting in reduced control performance. To address these challenges, this paper proposes Residual Koopman Model Predictive Control (RKMPC) framework. This method uses two linear MPC architecture to calculate control inputs: a Linear Model Predictive Control (LMPC) computes the baseline control input based on the vehicle kinematic model, and a neural network-based RKMPC calculates the compensation input. The final control command is obtained by adding these two components. This design preserves the reliability and interpretability of traditional mechanistic model while achieving performance optimization through residual modeling. This method has been validated on the Carsim-Matlab joint simulation platform and a physical 1:10 scale F1TENTH racing car. Experimental results show that RKMPC requires only 20% of the training data needed by traditional Koopman Model Predictive Control (KMPC) while delivering superior tracking performance. Compared to traditional LMPC, RKMPC reduces lateral error by 11.7%-22.1%, decreases heading error by 8.9%-15.8%, and improves front-wheel steering stability by up to 27.6%. The implementation code is available at: https://github.com/ZJU-DDRX/Residual Koopman.