I'm looking for a live chat application for a business to have one-on-one chat sessions with customers. Is anybody here able to point me to something that works? It will be on a Coldfusion server. Flash or html/ajax would be fine.

ColdFusion Enterprise comes with IM gateways ready to use. If you're interested in developing, check out:

http://www.adobe.com/devnet/coldfusion/articles/imgateway.html

If you're looking for premade, best bet might be to google around, I haven't used any of them to be able to vouch for any of them.