You can open the chat box by hitting enter, which places your cursor in the chat line, but by default it has no channel, so you'd need to specify what channel you want to type in.
However, the chat box will then close again after you hit enter to submit your message.
(Also you can steer your ship, fire blasters, and missiles with your mouse while the chat box is open, you just can't press the key binds for your power converter, invulnerability, or EMP while your cursor is focused on the chat window.)
A trick I've discovered to keep a chat window open during space combat is to have a second chat window tab, attached to your main window. After pressing enter to bring up the main window, click on the second tab and it will detach itself from the main window (while the main window then disappears). The second tab however will stick around (and reattach itself to the main window upon leaving combat). After a few minutes the tab may fade out, but, it will come back upon a mouse over or enter key press and stay for a few more minutes.