How do I successfully create a hook for an email override?

WordPress uses the PHPMailer class for sending mail. Before calling wp_mail, you can manipulate the $phpmailer object using the phpmailer_init filter.

//Place this inside of your form callback before wp_mail()
add_filter('phpmailer_init', 'foobar_phpmailer', 99999, 1);

function foobar_phpmailer(&$phpmailer){
    //Manipulate the phpmailer object here
}