Using the app, I've easily found the option to send a message to my opponent.
But on the website, whilst in the game and whilst it is my move, when I click Messages, then there is no option to type anything.
I'm probably missing something really obvious but would be grateful if someone would kindly point me in the right direction.

Above the submit move button, there is a "chat" icon. (This assumes a desktop view of the web site).