With the default Controllable setup, there is no binding for displaying the chat in multiplayer. It's possible to get it working by binding one of the buttons (e.g. right key on the D-pad) to the display chat action, then press the Steam button and X simultaneously to make the keyboard appear, then press the hamburger menu button to move the keyboard to the upper part of the screen so that you can see what you're typing, but this isn't particularly ergonomic.

It would be nice if I could just press one button to make the chat and keyboard appear, with the possibility of dismissing it without typing anything by pressing the same button.