In order to add terms to a taxonomy a user must have the correct permissions to do so. In WP for categories this would be
manage_categories. I doubt that there is any way to give this permission to non logged in users.
What you would need to do is set up some other type of field, a text field for example where they can enter either one new term or possible a comma separated list, or maybe a repeater. Then use an acf/save_post filter to get the new terms to be added and use WP set post terms to update the post https://codex.wordpress.org/Function_Reference/wp_set_post_terms
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