Solenoid to launch a ping pong ball
A solenoid could work. Another thing that could work is using compressed air. A ping pong ball in a tube that's slightly bigger than the ball's circumference with an electrically controlled valve could do it. You would basically want to replicate a t-shirt cannon on a small scale, and use paintball air tanks to get multiple shots before having to recharge. Changing the length of the tube would also let you have fairly good control over how far/fast the ball goes (longer tube = longer time that the force is applied to the ball and so greater acceleration). I think a solenoid would be a simpler option though. I would get some large ones and experiment with the repeatability and power needed to get the ball where you want to go.