To display Webshipper rates, third-party rates need to be enabled 

What is it?

To display your shipping rates directly in your checkout, you need to enable a feature in Shopify called "Third-Party Calculated Shipping Rates". This is a subscription limitation set by Shopify.

While the option is included by default in an Advanced Shopify subscription, you can contact Shopify support and instantly have it enabled for a smaller plan.

Do I already have it?

Check if you have it enabled here: https://your-store-name.myshopify.com/admin/settings/shipping

If you see the shipping truck, it is not enabled and you will need to contact Shopify.
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Contact Shopify

​Important

You will need to re-install the Webshipper plugin, if it was installed 
before you activated the third-party rates on your Shopify account.