Your emergency contacts are users of Truelytics and need their email address to log in.
Unfortunately you cannot update the email address of your Emergency Contact in your Continuity Vault because that is the email address your Emergency Contact uses to log into Truelytics.
To change an Emergency Contact's email, it would be easiest to delete the contact from your Emergency Contact's page and re-adding them with the new address. They will receive a notice about the removal and a new invitation under the new email address.