In my experience, customers who sign up, demand immediate customer support, then cancel a few hours later because they didn't get it, are doing you a favor - imagine what a drag on your customer support efforts they'd be if they stayed!
My rule of thumb is customer support thru email should aim for 24h turnaround, except if they're paying for "premium support". Great if you answer in 5min because you were just at the helm, but you shouldn't set unreasonable expectations.
Live support is murkier - if you do have set it up and are offline, falling back to email that gets answered tomorrow does seem slow, just by the fact you've just anchored me to "live". In my opinion, don't do it (or hide the widget!) unless you can really provide near-realtime response (even if via outsourced 1st level support agents).