I check your back-end settings and I couldn’t find any settings that will trigger your contact form to redirect to the 404 page when you save the changes. There might be some configuration added in the wp-config.php or .htaccess files that cause the issue. With that said, I suggest that you copy your live to staging area to investigate further without breaking the live website when we do our testing. Please give us the following information in a Secure Note.
Staging WordPress Login URL
Staging Admin level username and password
Staging FTP access
You can find the Secure Note button at the bottom of your posts.