Hello,
The PHP Codes appear to be fine. The issue appears to be with the Hosting Environment. Can you please update the reCaptcha Keys are simply remove it, so that we can check your Website online and determine what is causing the issue. This way we can provide you with more accurate solutions.
Hope this Helps!
Additionally, you can also consider following the steps below:
Firstly, please make sure that your PHP Version is more than 5.6 or preferably the Latest Version.
Secondly, it might be possible that the PHP mail() Function is blocked by your Web Hosting Provider in order to to prevent SPAM. So, it is Highly Recommended to use SMTP Functionality: http://docs.semicolonweb.com/docs/forms/form-setup/#docs-snippet-php for Forms as it ensures that the Mails are delivered securely.
Please check these 2 issues above and if you are still facing further issues, then please consider adding the .customjs Class to the .form-widget Element and then following code at the very top of your include/form.php File:
[ch_pre type="php"]error_reporting(E_ALL);[/ch_pre]
which should display all the possible errors. Please notify us about the Error Message you see, after going through the process above, so that we can provide you with more assistance on this.
Let us know if we can help you with anything else or if you find any further issues.