There's an option (in the optiopns > controls menu) to focus on the text box when you open the communications menu. This has the effect of turning any standard keyboard input into text. Pressing Q and E just types "QE" in the box.
To get out of text input, hit "Esc" once. You wil then be able to navigate with Q and E, space and WASD. It's a bit fiddly.
You can change the behaviour of the comms panel by changing the option in the controls menu to not focus on the text box; you'll have to pres space for local/wing chat or navigate directly to a pleyer and select them for the focus to switch to the chat box.
It's a fiddly pain in the bottom.
Protip: <TAB> cycles through all your potential chat targets when the comms menu is open.