Integrate 3rd party plugin to wordpress website

The 3rd party API is Rest API belongs to the shipping carrier company and we need to implement it to the WordPress woocommerce website. The case is that the shipping carrier is giving different prices based on the location of the buyer. The API should be implemented for checkout pages so that when the buyer tries to buy a product from the website, in the checkout page based on API the location should be selected via map (or location service), and based on the selected location shipping cost should be added into the total order price. Moreover, information about the order should be sent to the shipping carrier (order number, product, buyer location, etc). Provide time and the cost for implementing this program.