I have taken a look through ourShopify App Storeand have found a lack of apps that can integrate NZPOSTshipping rates at the check out. I can definitely put this through as feedback on your behalf to our developers for consideration as this would be very beneficial to have on the platform.
Regarding theCarrier Calculated Shipping, this is included in your annual fee. In order to have this activated, you'll need to get in touch with ourSupport Advisorson live chat/callbacks so they can access your store information and enable CCS on their end.
While you spend time working on your Shipping section of your website, I would like to include a helpfulShopify Blogbelow for you to read as it includes great tips on improving yourShipping Strategy. With the right shipping strategy in place, you'll ensure customers have a positive experience thus contributing to factors such as traffic and conversion.
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As Nia has mentioned here, I'm not sure if there are any apps that integrates with NZ POST. However, if you are open to using other carriers like FedEx, UPS, TNT, Aramex, DHL or any other, you can checkout the Multi Carrier Shipping Label that helps you automate the shipping & tracking process.
And yes, the carrier calculated shipping feature is included in your yearly subscription. You just have to contact Shopify support to enable it in your store.