Programming WordPress to Send an Email on Registration Form Submit?

While you might be able to do something on the client with jQuery I think your best bet would be to use the Post/Redirect/Get Pattern and submit your registration form via a <form method='post'> action.

The code after the screenshot below is a “Page Template“ for a “Page“ illustrates how to achieve what you’ve asked for with WordPress. Copy the code to your theme directory as page-contact-form.php (or some other name you like better) and then create a page that has template set to “Contact Form” as you see in the screenshot (note I kept the page template as simple as possible to illustrate the technique; you’ll typically have a lot more code in your real live page templates):

_{(source: mikeschinkel.com)}

<?php
/*
  Template Name: Contact Form
*/
define('ADMIN_USER','mikeschinkel');
if (count($_POST)) {
  $admin_user = get_userdatabylogin(ADMIN_USER);
  $registered_email = $_POST['email'];
  wp_mail($admin_user->user_email,
    'New Site Registration',
    "New Site Registration:\n\tEmail: {$registered_email}.");
}
?>
<?php get_header(); ?>
<?php if (count($_POST)): ?>
    <p>Hey <?php echo $_POST['email']; ?>, thanks for registering!</p>
<?php else: ?>
  <form method="post">
    Email: <input type="text" name="email" />
    <input type="submit" value="Register" />
  </form>
<?php endif; ?>
<?php get_footer(); ?>

After you’ve created your page with the above page template it might look something like this on your external site:

_{(source: mikeschinkel.com)}