Hi, this is a simple fix. This is not necessarily a Contact Form 7 issue. in fact, if you were to use something else like Gravity Forms, or any other form mail system, you'd have the same issue. The problem is that mail is not configured on the GoDaddy and that is fine.
What I'd do is configure your WordPress mail to work with a 3rd party free to use mailing system, which will also tell you if your mails are being received, bounced, opened etc... We use this system on all of our WordPress websites and it's fantastic. It would take me about an hour to get this taken care of for you.
I'm available between 6pm and 11pm Easter time. Freelancer.com is a side job for me, I work as a full time developer (WordPress, Laravel - PHP).