Sending Emails from Contact Form 7 works flawlessly, without any effort, for most people — but not for everyone.

Though the plugin successfully sends millions of emails every day, there are a host of issues that can delay or stop emails on both the sending and receiving ends of a typical email.
The important point to note is that the sending and receiving of emails depends heavily on both the server where your WordPress installation is hosted and the email configuration of the local computer where the emails are supposed to be received.

There is an extensive article written about possible flaws, which prevent sending emails here:

Please read through it and it will help you to identify the issue.