Spin two wheels in opposite directions with single DC motor?
You can connect the two with gears, or, since the ping pong balls are light weight, just find a pair of wheels with a diameter about 40mm greater than your current wheels and let the friction play the role of gear teeth. Four wheels would reduce the load your bearings need to apply to keep the shafts aligned; just put the small wheels in the middle.



Alternatively, you could just forgo the second axle altogether, and spin the ball against a static plate.  It would have a lot of spin when launched, but that might be useful, especially if you can get backspin on it so it curves upwards in flight.