Add a form between shipping method selection and order review
The payment panes makes this kind of interaction possible through a "details form" that open when you select the payment method that has this configured, and this is great because everything happens on the same page via ajax. But i don't know if Commerce Shipping has this too.
If not, i think that the best way for you to make this is to create a Checkout Page specific for your Shipping Method and create a Checkout Pane with your form to be in this page.

You can take a look at how https://drupal.org/project/commerce_correios handle this situation, because it's a Shipping Method too, so the API should have a lot in common with what you need.