Active front steering (AFS) can enhance the vehicle yaw stability. However, the control of vehicle yaw rate is very challenging due to (1) the unmodelled nonlinearity and uncertainties in vehicle dynamics; (2) timely response in control scheme. These two issues can be simultaneously alleviated through a random projection neural network (RPNN) for its high model generalisation and fast computational speed. However, typical RPNN cannot be directly applied to adaptive control applications. Therefore, a new RPNN-based adaptive neural control method is proposed, which is equipped with a newly designed adaptation law based on the theorem of Lyapunov stability. To test the performance of the proposed control method, simulations were carried out using a validated vehicle model. The simulation results show that, compared to conventional backpropagation neural network (BPNN) based controller, the proposed RPNN-based adaptive controller can reduce the response time and attenuate oscillatory steering in the case of cornering manoeuvre under fast variant vehicle speed. The results also demonstrate that the proposed RPNN-based adaptive controller outperforms the state-of-the-art fuzzy logic controller and the error feedback controller in multiple aspects including tracking nominal vehicle yaw rate, desired sideslip angle and intended path, showing its significance in vehicle yaw stability control.