Contact Form 7 4.8 and later versions use the WordPress REST API for Ajax submission. If the REST API isn’t available on your site, you have to give up using Ajax submission. You can still use non-Ajax submission. To deactivate Ajax submission and use non-Ajax submission, stop loading the JavaScript on pages on your site.

You may want to reevaluate the deactivation of REST API. REST API is one of the most fundamental functions of current WordPress. All security issues reported in the past have been resolved. Consider reactivating it.