Looking through the widgets on my WP.com blog and I found a widget for tlk.io which describes itself as a simple web chat. There's a plugin in the WordPress repository for use on a self-hosted site, but the plugin hasn't been updated for a couple of years so might be flaky.
Create new chat rooms in seconds. No invite or signup necessary. — Simple!
Privacy is yours
You can use a nickname without login. Or use Twitter to get your name and avatar. — Simple!
Easy to share
Share the URL anywhere you like. No need in forcing people to install the right app for just a quick chat. — Simple!
On the go
Chat in a browser tab at your desk or take it with you on your tablet or phone. You can even embed a chat on your own site. — Simple!