Similar to the WhatsApp Chat Widget except my website users can chat with other people, essentially creating a chat website.
Greetings and welcome to the Community, @Daniel8 
Thanks for sharing your thoughts with us!
You just need to let your visitors chat right on your website, correct? If yes, please upvote this idea - Live Chat App.
However, if I misunderstood your suggestion, please describe your use case in more detail 
I don’t really understand this Live Chat App. But basically I am trying to plan and see if there is a possible way to create my own chat app using Elfsight like Whatsapp.
The Whatsapp Chat widget works as a button that redirect users to WhatsApp app/web version.
Currently we don’t have an app that will let your clients chat with you right on your website, but we have a Live Chat request for this.
We also have a Group Chat idea on the Wishlist - this app should let your clients talk to other visitors on your website. If this request covers your needs, feel free to upvote it
Thank you so much