This happens frequently.
If updating the plugin and / or Wordpress is not solved, the problem may be one of these two things:
The domain from which the email is sent is not the domain of the site. This is the "from" in Contact form 7. The "from" field or the email that will appear as the one that sends the Messages must have the same domain as the website. So if the site is www.misitio.com the email it sends must be example @ mysitio.com (it can not be gmail, or hotmail , or any other).
This is because some hosting prevent the sending of mail from domains other than those of the site itself to prevent the sending of spam.
Generally if the field "from" has correctly placed an email with the domain of the same site but still does not send then it is for the following reason and the following should be done.
The hosting disabled the mail () function to prevent the sending of spam. Then the emails must be sent through SMTP. For that you must install the WP Mail SMTP plugin and configure it with a real email account.
The plugin link: link
Here's how to configure it: link