Short and simple explanation: id provide the embed code for the live chat box that features on the customers website, my web app would handle displaying the received messages to the customer support team and letting them respond, bringing live chat to smaller companies who can't develop their own in house solution
Simple and understand what you're trying to do @Thespragg good step!
What's your plan to execute differently than existing solution like Tidio, Jivo, Intercom and Drift?
I use Zoho, their free package is also good.
We use https://crisp.chat/en/ free package it is very good.