There are hundreds if not thousands of fulfillment companies. There are a few very well known companies, but there are also hundreds of local fulfillment companies.
The bigger companies that have direct integrations with WooCommerce should automatically retrieve and update order data, but, if you're working with a smaller company, it's important to know what you have to do to send them data and to update data on your end.
We're going to be looking into how you send shipping information to any company and how you can update your own shipping data.