You can embed Slack into a website using a 3rd party tool called Taggbox Widget (free for 2 feeds)
Discord natively supports embedding with its own widget.
In my opinion MS Teams will eventually eat Slack and Discord thanks to easy corporate IT service bundling, so if you’re a glutton for punishment you can apparently embed teams chat with a 3rd party app from the Teams App Store
I’ll update with other potential embeds as I find them.