If you are developing a Shopify App, you have two options to deliver a java script to your app's clients store:
1. Using ScriptTags - recommended
2. Saving the js to the client's theme assets using the Asset API.
I can't see any image attached to your post.
However, if you save the js in the theme's assets folder, it will be hosted on cdn.shopify.com.
If you could post more details, like the "browser notification" you are referring to, I might be able to help.
Hi @alexsvderV ,
That popup looks like a confirmation popup for allowing push notifications from a site.
You cannot send push notifications to a client without his approval.
You can find more details here.