A non-Slack alternative: I was looking for this feature, but wanted to move my company away from Slack to one of the free chat platforms like Rocket.Chat, Mattermost, Zulip, etc. So I researched, and found that only Rocket.Chat has its own version of this live-chat feature (though no doubt much less mature). I'm looking forward to switching over to Rocket.Chat soon (I have no affiliation with them).
Hope this helps folks out there looking for live-chat integration options. It's pretty cool to be able to chat with potential users directly from company chat!