E-commerce sites are more and more commonly offering live chat for their visitors as a way of customer support. There is a wide selection of modules for Drupal that can add this functionality. However, most, if not all, I found rely on third party chat services. A while ago, I decided to build a native live chat module in connection with a project of mine. The module has been released as Customer chat.
The primary interface of Customer chat module is a dashboard that site agents can use to accept incoming chats from visitors. This page shows all the active chats that are relevant to the logged-in agent, each of which is tucked into a tab. Using those, agents can respond to or reject chat requests.
On the front-end, the module provides a block that shows a link for visitors to initiate a chat whenever an agent is connected. Visitors can start a chat session by filling in a form with their contact information. Once the chat request has been submitted, it will appear on the agent's dashboard, waiting for them to respond.
If you work with sites that could take advantage of a customer support tool, take a look at Customer chat. Feedback is always welcome - feel free to open an issue on the module's project page if you have any questions or feedback.