Sending SMS will be more practical. The operator is not near his desk to read email anyway.
However, when the operator is not around the desk, he may simply switch his status to offline mode in the monitor window. Then when the website visitor clicks on the chat request icon, he will be presented with a form to fill. This form will be sent as email.
I have connected my email to be forwarded to SMS service :-) So I need that upon chat request HCL send email to specific address.