This plugin appears to be broken when installed along with WooCommerce. WooCommerce uses the select2 library, it looks like ACF Font Awesome picker is trying to use the select2 library but a different (newer?) version has been initialized already by WooCommerce. I can reproduce the same issues on multiple sites with the latest version of WooCommerce, ACF, and ACF Font Awesome installed.
Note: I experience the same thing, no icons in the dropdown to select a default icon on the admin ‘edit custom field’ page, and when I update the icon on a post the value doesn’t save when the page reloads.