We have a ticket field called "Company ID (ticket)" that is autopopulated by the system when the ticket is created. We have another custom organization field called "Company ID (org)".
We would like for this "Company ID (org)" field to be autopopulated with the value from "Company ID (ticket)", when "Company ID (org)" is not Present. I have tried to do this with webhooks but I am stuck trying to change the org custom field helper with that.
Has anyone here done something similar? Is it possible to populate organization fields with webhooks?

I was able to update organization fields as in your example/scenario, please see the attached screenshot as example.
Further clarification is needed in your use case, what type of the fields "org_1" and "org_2" (text, milti-line or drop-down) is used?
I'm going to open a support ticket with you about your issue, where we can collobarate further.