"You do not have permission to view this page"
I suspect that there is no default page in your Azure website, which cause the issue. You can add/specify a web page as the default document.
Same issue appears if no default document (web page) in my project folder:

I've customized my web chat by cloning from the source here. Now I want to deploy this to Azure and host it on one of my websites.
You can follow the following steps to customize&build your own Web Chat, and then add it to your website.
1) Refer to the steps described in “Building Web Chat” to build your own Web Chat
2) Include your own customized botchat.css
and botchat.js
in your website project
3) Add references to botchat.css
and botchat.js
in the section of the web page
4) Add and initiate BotChat with directLine_secret
, botid
etc information
<script>
BotChat.App({
directLine: { secret: "{directline_secret}" },
user: { id: 'You' },
bot: { id: '{bot_id}' },
resize: 'detect'
}, document.getElementById("bot"));
</script>
5) Test it on local, if WebChat works as expected then publish the website to Azure
I did a test with the above steps and it works fine both on local and Azure website.

When I try doing the same on Azure, the chat never loads and shows a blank window.
You can check if botchat.css and botchat.js are loaded as expected via F12 developer tools Network tab, and you can check if any error appear in developer tools Console tab. – Fei Han