Need to know what sort of chat you are really requesting. Primarily, will it be available between visitors? Or only between a visitor and yourself? Will there be multiple channels possible at one time? That is, if the chat is only between yourself and a visitor, can there multiple such sessions at one time?
Live chat in essence is simple; but making a real chat with the "usual and customary" accessories (what people expect form a chat) is more time consuming: formatting, emphasis, smiley faces, a chat stream (pretty essential), archiving of previous conversations and so on can get mildly complicated.
So, you see, right now I have more questions than answers, but on the other hand I can do a chat system, just need to know the specs. How much of that falls with in your budget is dependent on what you are asking for.
I've been a programmer for a long time. I have worked remotely with clients for a long time, also.
Let me know if may be of service to you,
Phil -- Computer Whisperer