I suggest you first of all to separate code from markup, especially dividing your executing file like the function support_form().
Next, please try to send email without form with hard coded message. It is possible you have mis-configuration on your server if you developing on local machine.
And finally, you should fill action attribute for <form> tag. And that must refer to the executable script. Otherwise it tries to execute you homepage where this code is absent. If you are not planning to separate your markup and code I suppose you should link your action to the file that contains this script. When that happens, your code
if ( isset($_POST["submit"])) {
do_action('my_phpmailer_example');
}
will execute.