This is a common issue with Contact Form 7 and some php mail or server settings on some hosts.
Try hardcoding the sender name in the ‘From:’ field in the ‘Mail’ section like Webmail <a-valid-address@mydomain.com>
This means you won't see the sender name or email as names and return emails in your incoming mail box, but that doesn't matter much, as the sender's email will be in the body of the message.
If that doesn't work, try https://wordpress.org/plugins/wp-mail-smtp/ to use SMTP instead of php mail.
And see http://contactform7.com/faq/ and http://buzztone.com.au/contact-form-7-email-issues/