Square keeps the online store and the in-person register in sync with each other by default. It doesn't extend that same sync to a separate warehouse system, a third-party marketplace, or an accounting platform, any of which still needs its own connection built if the business sells or tracks inventory somewhere Square itself doesn't reach.
An online order also still needs to become a real fulfillment task somewhere, picked, packed, and shipped, which Square's dashboard will show you but won't act on by itself. For a business running any real volume through the online store, someone's either watching that dashboard constantly or something's watching it for them.