I’m having the same problem.
Mysteriously, Contact Form 7 worked fine for a while, seemingly compatible with all my other plugins.
…Then it mysteriously stopped working. For something as essential as a contact form, I recommend NOT using Contact Form 7.
A site’s contact form is crucial. You can’t just let us down like that.
Past threads have advised dumping Contact Form 7 and going with Jetpack instead. That just might be the best solution.