How are widgets, such as Crisp, created?

Question

Vue.js

Hello community!
The task is to make an external plugin like Crisp-a, namely what is and what is needed at the output:
There are 2 servers: Server
B - Backend. It hosts a js server that works with the plugin over a socket. It hosts the plugin itself.
ServerC - client server where my plugin (chatbox.min.js) should be connected.
On the ServerC client server, there should be a simple plug-in connection via

<a><i class="icon-jingle" /></a>
<script src="https://СерверB/chatbox.min.js">

By clicking on the icon, a chat should open in a modal window, which will communicate with ServerB via socket.io.

I'm currently reading Flanagan's Javascript. Everything is scattered here. Dear professionals, can you tell me about technologies/libraries? How do I program, via es6 Classes? How then to put everything together? How to add html code to plugin? Is it realistic to make it reactive with VueJs?
In general, help collect the vermicelli in your head in one pile.