if I wanted to change the word "Free" rate_formatted in the Delivery Methods in the checkout, Can this be done in Extensibility API?

I don't fully understand what you want to do, but if you just want to change what text is shown for a string, you can add a translation to the site's domain configuration record.

Hi Steve, Like changing the Free to zero in the template

Right, well to answer your original question in a simple way - that's not something you can do with the extensibility API. You would likely need to change the JavaScript.