hook into third party API after checkout
You should probably be looking to use the hook_uc_checkout_complete() hook.

From the docs, it takes two parameters, $order and $account - I'm guessing $account will be the one you'll need to create the account on the third party web service as it's the object that contains the customer/user information.

The registration part of your question depends completely on which third-party web service you are talking about, and will probably fall outside the scope of Drupal Answers unless there is either an existing integration module, or you are looking to write one and need help on the Drupal side of things (i.e. this is not the place for general API usage questions).