I'm going to assume when you say saving the chats on the client window you're referencing the chat widget? So when a customer returns to the website they can resume a chat?
If so: Crisp.chat, Kayako, Intercom, LiveChat.com is going this route, HelpScout to name a few.
Most likely these would store a cookie so if the customer is returning with the cookie in place then they'd be able to resume chats easily. This can be taken a little further with most if not all offering the relevant code to associate a logged in user with a customer in the chat app.
Alex, that makes much more sense with regards to the storage.
Somehow I read the post that the user was looking for the chat session to be stored on the client's end of things. Re-reading the post now, they were just looking to have the chat window restored, and that can be done with any of the softwares you listed.
I used to use Kayako (started with them when it was $99 for a lifetime license), and I use Intercom with a specific host that our clients use. I've not heard/used the others that I recall.