Can Locksmith lock Shopify's public JSON API for my online store?

It cannot. To our knowledge, this feature cannot be disabled, and its behavior cannot be adjusted – and, it's an integral part of the operation of many themes. If you have questions about this, contact Shopify support.

If preventing access to these endpoints is mission critical, you could consider building a web proxy that allows all traffic except those endpoints. This would be a custom project, and it's not something we can help with, or support.