I don't believe this is possible right now.
I created an organization and added a coworker to it. She created a board within the organization. I went to the board and opened up its preferences and saw:
In the screenshot, creating invitations to the board is limited to members of the board. Furthermore, you can see at the bottom of the screenshot that you must be an owner of the board to set the preferences related to creating invitations.
Organization owners can set it so anyone in the organization can vote or comment on any board, if they want, but that freedom does not extend to creating invitations for boards.
You can consider requesting the feature here.