I'm not too sure about that. The
Title 10 article under which Trump ordered these guardsmen/women to active duty includes this statement:
"Orders for these purposes
shall be issued through the governors of the States or, in the case of the District of Columbia, through the commanding general of the National Guard of the District of Columbia."
While my experience with US Army pers admin was very limited and long ago, my impression then was it made similar CAF processes seem almost invisible. After a quick look at AR 600-8-105 (the regulation prescribing policies and mandated tasks regarding military orders) and DA Pam 600-8-105 (describes the orders formats and their use in specific circumstances), it appears that any of the formats that may apply to order guardsmen to active duty under that specific title mandates including a statement to the effect that the individual is ordered to active duty with the consent of the Governor. There are formats that do not include a "Governor's consent" but they are for different circumstances.
The Governor of California has been very clear that he has not consented to use of the National Guard. I don't doubt that documenting the duty status of the guardsmen will eventually be accomplished. I am less confident that "those papers will be sent to their units within a day or so", especially since it's their units' responsibility to cut the orders.