Here is a reference point for the CarrierService API --> https://shopify.dev/docs/admin-api/rest/reference/shipping-and-fulfillment/carrierservice. Here is how you can install an API credential set against the Shopify store --> https://shopify.dev/tutorials/authenticate-with-oauth. Once you have the API credential set then the CarrierService API should be available if the OAuth authorization included the write_shipping access scope. As long as your Shopify store has the API credential set active so that the CarrierService API calls will be valid then the third party can plug into that.
Hope this helps some at least!
If your store doesn't currently have access to the carrier service API, there's a few ways to get it. If you pay for your subscription yearly, you get this feature for free. Otherwise it can be added to your current plan for $20/mo, or you can upgrade to the Advanced Shopify plan which also includes the feature. More details in our doc here.