Sorry for such a brain dead question but we have people that want to register for our events that have no email address. Hard to imagine in this day and age but it happens. A lot of our contacts are old and not up with all the newfangled internet stuff. I have set up an event registration form that works very well for most, but requires email address. I have googled "civicrm register event email address" and not found anything specific on this issue.
Make email address field not-required on the profile you include on your event online registration form.
If the registering participant does not provide an email address, you could get duplicates in your CiviCRM contacts. See this warning:
Duplicate Matching Impossible The selected profiles do not contain the fields necessary to match registrations with existing contacts. This means all anonymous registrations will result in a new contact.