There are companies like DeskMoz (https://deskmoz.com) which are dealing greatly with live chat tool, They covered it completely by providing 24x7 live chat trained agents. But the main question arises when it comes to integration, There are lots of platform like WordPress, google analytics and various plugins which can be more beneficial to a company. What is the response to every live chat tool with integrations?
This said, all you really require... choose how you'd like the live chat icon/window/widget to display + likely the live chat provider will have a library of code to display their live chat window in different ways.
As per my experience we have been using Zendesk and didn't face any issues either with technicalities or support.