Rocket Chat allows you to self-host for free. At its core it is very similar to Slack plus it has video conference capabilities. You can install rocket.chat on your own server or in the cloud with AWS or Digital Ocean.
Scroll down to the server section at https://rocket.chat/install for installation instructions.
Once you get it going you can check out this page to learn how to get the script to embed into your website: https://rocket.chat/docs/administrator-guides/livechat/
When you are done you will have a chat icon in the bottom corner like on this site. You can setup the messages to go to active members on your Rocket Chat or even setup the messages to go to email during after hours. This can be a useful tool to also avoid integrating captchas and avoiding spam.