Can a landlord accept termination notice from a tenant by e-mail only, or does there also need to be a hard copy?

A landlord and tenant can agree on any form of notice they choose.  If the lease contains notice provisions, it's a good idea to put any later agreement about other forms of notice in writing.  Notices are generally considered given five days after they are mailed, two days after they are sent by FedEx or Express Mail or other overnight service, two days after the fax transmission ends when faxed, and same-day when hand delivered.  There is no clear rule for email and because it is subject to spam filters and other things that might delay or prevent delivery, it is hard to say when it would be effective unless the parties agree or receipt is confirmed somehow.

Generally, the manner in which the tenant may give notice is provided for in the lease under the Notices section.  If the notices section states that the tenant must give written notice, then email is an acceptable method unless specifically stated otherwise.  For instance, if notice must be given via certified mail, then email is not acceptable.   The date that the notice is dated is usually the date that controls when you count from.  If you are concerned that the tenant back dated the letter, then you may be able to check the date it was mailed by looking at the stamp.  As long as the notice simply says "in writing" the email is sufficient.