ACF 5 has the ability to allow new terms to be added to a taxonomy.
In order to allow the this for your users you’ll also probably need to adjust user permissions on the site to give their user type the ability to add terms. There are several user role editor plugins available on the WP repo.
Unfortunately, I’m working with ACF 4. If this is only on ACF 5, could you perhaps show me screenshots of this with a user admin please (how the settings look, how it looks on front-end, etc)? I would love to pay for Pro if I can be sure that it will be able to fill all the requirements for this project but until then, I’m stuck with ACF 4.