Say, is the chat edit box always supposed to be available in the chat window or do I have to interact with the chat icon each time? Sorry to say, I thought I remember seeing it without clicking on the chat icon; maybe my memory is faulty.
Usually it appears when I press / button to start chatting, but if you have the combat log open as well it may get confused with which chat frame it is in as the edit box switches its owner based on current chat frame selected. So if the / button doesn't work try clicking on the chat frame or toggling the combat frame to a different infopanel.