There are many options for this task. A few all in one services such as boldchat, egain or liveperson allow you to manage emails, chats and other site functions. With the email component, you can have the system sort the emails and have ownership of each email assigned to an individual employee.
I am not sure of any free products out there. We researched the above companies when we were looking for a new online chat and email response program.
I use open source web packages for this. Basically for each site I use a helpdesk where all emails are retrieved from the admin back end. A helpdesk has a ticket system so you group emails by tickets, you can assign them to different departments for management, setup priorities, search, filters etc. Being open source you can customize it in many ways and/or have other developers work with it. Same goes for live chat modules.