In my case, I had existing contact forms prior to upgrade to 4.8.1, I noticed that the REST API urls were returning 404 errors because the path became incorrect, it became
/wp-json/contact-form-7/v1contact-form-7/v1/contact-forms/[id]/feedback instead of /wp-json/contact-form-7/v1/contact-forms/[id]/feedback
Resaving all the forms corrected the path and fixed this issue for me
