Vehicle platoon driving can improve traffic capacity, fuel economy, and road safety for intelligent transportation systems. Motion control of vehicle platoons serves as the cornerstone for the realization of intelligent platooning systems, whereof the string stability should be first investigated distinct from single vehicle control stability. Existing literatures predominantly concern longitudinal string stability. This study presents a distributed controller that simultaneously addresses string stability of lateral and longitudinal controls. Wherein, a decoupled lateral-longitudinal control approach is adopted. Moreover, a hierarchical control architecture is employed for the longitudinal control, including a PI controller with anti-windup integration for the lower level and a PD controller for the upper level, optimized by using RBF neural networks. As for lateral control parts, a nonlinear controller is grounded in the Stanley steering kinematic model. Then, both lateral and longitudinal string stability conditions for vehicle platoons are theoretically derived. Finally, the proposed vehicle motion controller is verified by simulations of an electric vehicle platoon. Results indicate that the longitudinal platoon string stability meets design requirements, while the lateral platoon string stability exhibits velocity-dependent constraints.