In-built Creatio functionality allows integrating with Telegram and Facebook chats and the procedure is described here (I believe you used this article to integrate with Telegram in your case). You will need to study the implementation of the Telegram integration in the system (server code part, not the Creatio UI part) and check if the API of your site chat allows integrating it with 3rd party services.
Unfrotuantely there are no tutorials on the Academy on this particular topic since this is a server-side development using API services.
As for introducing the new tag - I will ask our community team to add one.
Does this method require Creatio to be on-site? We have it in the cloud currently
No, the endpoint where you chat logic is stored should be available for the cloud-based apps. If it has no firewalls and can send\receive requests\responses from the Internet then there is no difference if the app is located in the cloud or locally.