For your swift mailer include please check your pathing and ensure it is correctly pointing to the file you need.
else include_once "../Sendgrid/lib/swift_required.php"; // the user has submitted the form
For example if Sendgrid is a folder in your root directory, you wont need the .. it might only be the following:
else include_once "Sendgrid/lib/swift_required.php"; // the user has submitted the form
You also mentioned your running an Azure Website, something to take note of is Azure Websites have error messages turned off by default for security reasons(as they should) but if your site is still in development then you should turn custom errors on or you won't ever get a meaningful error message when things go wrong in your php code. Make sure to turn it off once your site is ready for production!
For Azure you can't directly turn on errors in the php.ini file since they restrict this but you have to put a custom .user.ini file in your root directory with the following
display_errors = On
This is why I suspect your site wasn't telling you anything is wrong. More info on .user.ini files here at this link: http://blogs.msdn.com/b/silverlining/archive/2012/07/10/configuring-php-in-windows-azure-websites-with-user-ini-files.aspx