Diaper services use far less energy than disposables or washing from home. It's an industrial process, so by not stopping the system, they don't have to use as much energy to keep the water at super hot temperatures.
The delivery service is actually fairly negligible, it's long efficient trips by one car, rather than the many inefficient trips to the grocery store by individual cars needed for disposables.
Washing from home with a high efficiency machine, might match a diaper service with delivery, but I don't know if that study has been done.