It's already possible to move while typing - hold a directional key (W, A, S, or D), then press your chat keybind and start typing. Your character will keep moving in the same direction as long as you hold the directional key, but you won't be able to change it nor will you be able to use the directional key in your message (since you're holding it, and releasing it brings your character to a stop). While not ideal, this still comes in handy when chasing people or generally avoiding having to stop to type while walking.