80.19 (F)(1) Upon completion of any installation of electrical equipment that has been made under a permit other than an annual permit, it SHALL BE the duty of the person, firm or corporation making the installation to notify the electrical inspector having jurisdiction
It could easily be argued that the GC is the firm or corporation making the installation even if he hired the work done.

Can the owner or secretary of Sparks Electric call or does their exact employee that "made the installation" have to be the one that calls.

GC is responsible for the whole job. It's silly to not let him call for inspections whether the individual trades are available or not.