Serve a javascript app with templates

I'd like to develop a plugin in javascript, which will be served on another website the simpliest way possible.

The idea is to display a chat popup usable by the visitor of the website but entirely scripted on my side. I don't have trouble developing it locally but I have no idea how to implement it on a client website.

Is it possible to have something this way :

On client website :

<script src="http://ift.tt/1oGBy3t"></script>
<div id="myChatPopup></div>

With the script using the div to implement the templates. Ideally, I'd like to use VueJS. Is there any webpack plugin to package a javascript app this way ?

Thanks.