We have logged a feature request in our feedback portal for implementing a built-in chat module. Hopefully we will be able to provide this functionality in our future releases. Here is the link
to our feedback portal where you can track the progress of the feature request and vote for it.
Currently, you can use a third party chat application and implement it to work with Sitefinity as it would with any standard .net application.
Do you want to have your say in the Sitefinity development roadmap? Do
you want to know when a feature you requested is added or when a bug
fixed? Explore the Telerik Sitefinity CMS Ideas&Feedback Portal
and vote to affect the priority of the items