Adding terms to a taxonomy requires the WP manage_categories permission.
Your choices are limited.
You could edit roles and give everyone this permission, but that’s probably not a good choice.
I might be possible to give this permission temporarily when viewing a specific page. I’m now 100% sure about this or how it can be done.
And the last choice is to include a separate text field that is only shown when a user does not have the correct permission using the acf/prepare_field filter and then adding an acf/save_post filter that will add the term to WP and then set it as a post term.
