I'm thinking about creating the chat functionality from scratch in order to provide other companies that don't know how to start with it.
In general there are already some chat plugins available in the market. Ref:
https://themeisle.com/blog/best-live-chat-plugins-wordpress/ or search on google by typing "wordpress chat plugin".
Instead of developing a script from the scratch, if I were you then I would use a ready-made one first. I would the customize the existing one to meet my needs. Hope it helps.
Good luck