If an agent responds to more requests than others, is there a way to auto-route chats to other agents?
By default, every Chat account has Broadcast as the selected routing option. This means that all agents are notified when there is an incoming request and they have to serve the request themselves.
You can change the routing option to Assigned which routes incoming requests to an online agent who has been available for the longest time.
For more information about this subject, see this article: Setting up notification routing for live chat and messaging. For information routing tickets evenly among agents, see this article: Can Zendesk Support equally distribute tickets amongst agents or groups?