Add additional button to login form
You cannot alter a form from a theme's template.php file. You need to create a custom module for that. In this module, you can alter the login form in an implementation of the hook_form_alter() or hook_form_FORM_ID_alter() hooks.

Without some JavaScript, you will not be able to submit the form to a separated URL. What you can do, in your module, is to provide an dedicated validation or submit handler for you button. In this handler, you can redirect to your external URL using drupal_goto().

Something like:

MODULE_form_user_login_alter(&amp;$form, &amp;$form_state) {
  $form['foo'] = array(
    '#type' =&gt; 'button',
    '#value' =&gt; t('Foo'),
    '#submit' =&gt; array('MODULE_user_login_foo_submit'),
    '#limit_validation_errors' =&gt; array(),
  );
}

MODULE_user_login_foo_submit($form, &amp;$form_state) {
  drupal_goto('/path/to/request.php', array('external' =&gt; TRUE));
}