How can I implement a form into a DC view footer?
You might want to consider using the Commerce Shipping module to handle the integration to Drupal Commerce, it doesn't support adding the shipping quote form in the cart footer, but it would help in handling the actually shipping quote with commerce.

The problem you are facing is in the hook_views_data, you should do something like this:

function MY_MODULE_views_data() {
  $data = array();
  // Define an footer handler for displaying shipping quotes.
  $data['MY_MODULE']['shipping_footer'] = array(
    'title' =&gt; t('Shipping API Quotes'),
    'help' =&gt; t('Display a user input form to allow the retrieval of shipping quotes.'),
    'area' =&gt; array(
      'handler' =&gt; 'MY_MODULE_shipping_form',
    ),
  );

  return $data;
}


You need to define an area handler as shown above, that should let you select the form in the views UI.