How do I apply a wordpress function to something with no ID inside an archive page?

Your rating plugin will not work on custom HTML content (e.g. those boxes). Why? Because where will it save the rating related data? It must save the data somewhere, right?

Now, you don’t want to add the rating in any post, that’s fine. Your other option is to create a custom post type. You can do that using a plugin such as Custom Post Type UI.

WordPress allows you to create custom content (for example: Rating Box) known as custom post type, that can be used for any purpose.

Once you create a custom post type (using the plugin or your own custom CODE) and name it (for example) Rating Box, you’ll be able to use ACF plugin to create a custom rating field for Rating Box. Then, use this custom post type to develop your custom archive page. Once you do that, you’ll be able to rate them using the rating plugin (if that plugin supports custom post type – a standard plugin should support custom post type).

Another option will be to use the Settings API to save the data and show the rating on your custom HTML box using the values set from the Settings API or Options API.