Robot skateboarding is a novel and challenging task for legged robots. Accurately modeling the dynamics of dual floating bases and developing effective planning and control methods present significant complexities in accomplishing skateboarding behavior. This paper focuses on enabling the quadrupedal platform CyberDog2 to achieve dynamic balancing and acceleration on a skateboard. An optimization-based control pipeline is developed through careful derivation of the system’s equations of motion, considering both the robot and skateboard dynamics. By accounting for system physical constraints, an advanced offline trajectory optimization method is employed to generate various acceleration trajectories, creating a motion library for the system. An online linear model predictive control with whole body control framework is used to track the generated trajectories and stablize the system in real-time. To validate its effectiveness, we conducted experiments in various scenarios. The quadrupedal robot successfully performed acceleration from a static state to various velocities and demonstrated the ability to balance and steer the skateboard.