I know sometimes people receive their babies on Shabbos, but I can’t find any sources discussing the halacha of having either a stork do work for you, or benefiting from a drone, on Shabbos. Where did you learn from?
So it’s true that you can’t specifically request for the stork or the drone to drop off the baby on Shabbos. However, this is not an issue because you are paying the baby factory (and its messengers) for a job, not for specific hours worked, and if they choose of their own free will to deliver the baby on Shabbos, that is their own choice. It’s similar to how you can mail a package that will be in transit over Shabbos, because you are paying the post office to deliver the package, and it’s their choice if they choose to transport it on Shabbos vs. another day. Also, most people would rather their baby not be dropped off on Shabbos since it’s more complicated that way, so it is arguable that one does not actually benefit from the stork or drone anyway in such a case.