If you let messages go to email, do they come through with an email address you can respond to? I'd imagine that email replies that begin "I'm the founder of X" probably get a better-than-average open rate.

At the same time, I wouldn't want to frustrate people who typed something into what appeared to be a live chat window — expecting a live human — and got either a bot or a delayed email response (after a request for email address, of course). Seems like there's definitely upside potential here, but also the risk of frustrating/angering people. I'd hope there's a way to hide the chat box during certain hours, to mitigate the latter.