You can get your API key by going to admin > apps > bespoke shipping > settings (scroll to bottom)

It’s a HTTPS call to https://parcel-intelligence.com/cs/api_ex.php, the payload is similar to Shopify’s carrier service API (see: https://help.shopify.com/api/reference/shipping_and_fulfillment/carrierservice)

You will also need to set the 3 headers
Content-type: application/json
HTTP_X_STORE_NAME: xxxxx.myshopify.com <-- this is your store name in the format xxx.myshopify.com
HTTP_X_STORE_TOKEN: xxxxxxxxxxx <-- this is your API key
HTTP_X_SHOW_DEBUG: true <-- set this header if you want to see the output of print_r or var_dump functions, omit if you do not want to see it, also omit in production otherwise your json may not parse correctly.

The response and payload is in JSON

Example request
{"origin":{"country":"US","postal_code":"91210","province":"CA","city":"Glendale","name":null,"address1":"1 main st","address2":"","address3":null,"phone":"","fax":null,"address_type":null,"company_name":"Bespoke Shipping"},"destination":{"country":"US","postal_code":"10001","province":"NY","city":"New York","name":"John Doe","address1":"123 Main St","address2":"","address3":null,"phone":"9876543210","fax":null,"address_type":null,"company_name":""},"items":[{"name":"Product A - 250gr","sku":"","quantity":20,"grams":300,"price":1400,"vendor":"ACME","requires_shipping":true,"taxable":true,"fulfillment_service":"manual","properties":null,"product_id":9062306182,"variant_id":33082410182}],"currency":"USD"}