You don't need letter buttons. You can have a regular text box in an iframe and submit the form with the text.

I actually remember websites doing exactly that years ago.

Some would display the chat like here, by never closing the connection, as described in the repo. And another trick was to just send a refresh header (or use a meta tag) to automatically reload the iframe displaying the chat every $x seconds. The latter one was iirc more prominent because you could do it on any web host with PHP and MySQL as it doesn't require any long living processes.

That's exactly how I implemented a web chat around 2002. IIRC, Internet Explorer didn't update the iframe with the chat messages reliably, so I had to fall back to reloading. The whole thing was rather inefficient, simply polling a database for new messages every second, but it worked fine for about 10-20 users.