Hi, I have had the same problem and the rootcouse was the disabled WP REST API. As I read you have installed WP REST API plugin that disable the WP REST API which is used by the Contact Form 7. Try to enable it and check the functionallity again.
(WP Rest API could be disabled from the functions.php file also. You can disable only the endpoints (e.g. users), and not the entire WP Rest API.)
