I don't really think you know what you are asking here... A chatbox would need far more work than most people would do for some stranger on the internet. You are going to need a server to host your chat. The only alternative I can think of would be using IRC. I will try to create a small irc widget for you, but remember: anyone can join the chatroom so privacy will be limited.
https://greasyfork.org/en/scripts/13921-greasyirc To make it work on your unspecified site, edit the script and change the @include to htt://site.com/*
I'll buy an easy script which can include a chatbox on a target site.
Hello, i would like to chat to my friends on a gaming site, we are a team and we don't like to chat public there. I'd like to chat to my friends on a custom private chatbox, which can only be connected by having the script, and a focussing to chat button. That's all. I can pay some dollars if it is useful and how i wanted