Check your Net panel in Chrome or Firebug. I am noticing that there are two ajax requests going on.
That is, there are two requests, one for REFILL and one for FEEDBACK.
REFILL seems to happen immediately on page load, FEEDBACK when the form is submitted.
I am using the Straker language plugin and it puts a language prefix in front of every URL, so this is what I get:
/es-mx/wp-json/contact-form-7/v1/contact-forms/130/refill
/es-mx/wp-json//contact-form-7/v1/contact-forms/130/feedback
By adding an .htaccess rule like this:
Redirect /es-mx/wp-json/contact-form-7/v1/contact-forms/130/refill /wp-json/contact-form-7/v1/contact-forms/130/refill
Redirect /es-mx/wp-json//contact-form-7/v1/contact-forms/130/feedback /wp-json/contact-form-7/v1/contact-forms/130/feedback
Essentially, because I am using Spanish-Mexico (es-mx) as a language prefix, then the REFILL request needs to redirect to something NOT prefixed by es-mx.
I am able to solve the REFILL not working but the FEEDBACK is still a problem.
I suspect that the FEEDBACK url that goes out is WRONG, as it has two slashes there after the wp-json path.