I had this problem recently on this site that was really baking my noodle. It turned out to be a mixture of SELinux and reCAPTCHA misconfiguration.
First, check if SELinux allows the web server to send emails from your site:
$ su -
# getsebool httpd_can_sendmail
If the boolean is off, turn it permanently on:
# setsebool -P httpd_can_sendmail 1