Wordpress - wp_mail script with jQuery post

First add your email processing function and hook it to wp_ajax hooks like this using your functions.php:

// if you want only logged in users to access this function use this hook
add_action('wp_ajax_mail_before_submit', 'mycustomtheme_send_mail_before_submit');

// if you want none logged in users to access this function use this hook
add_action('wp_ajax_nopriv_mail_before_submit', 'mycustomtheme_send_mail_before_submit');

// if you want both logged in and anonymous users to get the emails, use both hooks above

function mycustomtheme_send_mail_before_submit(){
    check_ajax_referer('my_email_ajax_nonce');
    if ( isset($_POST['action']) && $_POST['action'] == "mail_before_submit" ){

    //send email  wp_mail( $to, $subject, $message, $headers, $attachments ); ex:
        wp_mail($_POST['toemail'],'this is the email subject line','email message body');
        echo 'email sent';
        die();
    }
    echo 'error';
    die();
}

Then inside an your theme's js file create the AJAX call like this:

jQuery('#donationForm').submit(function() {

// send email to client before moving onto worldpay payment form
var data = {
    action: 'mail_before_submit',
    toemail: $('#myemailfield').val(), // change this to the email field on your form
    _ajax_nonce: $('#my_email_ajax_nonce').data('nonce'),
};
jQuery.post(window.location.origin + "/wp-admin/admin-ajax.php", data, function(response) {
    console.log('Got this from the server: ' + response);
});

});

Now add the nonce to your footer.php since it needs to be generated via PHP:

...
<span id="my_email_ajax_nonce" data-nonce="<?php echo wp_create_nonce( 'my_email_ajax_nonce' ); ?>"></span>
<?php wp_footer(); ?>
...

And you should be set.

Basically, you can use JavaScript to post to a WordPress function, then the WordPress function can invoke wp_mail() to send the message.

A great place to start would be the useful AJAX in Plugins article on the Codex. It walks you through all the steps required to add your JavaScript on the front-end, your PHP on the back-end, and everything you need to do to tie the two together.