Can customers export a list of their tickets from the portal?

Yes, there is an option to export tickets in the customer portal as shown in the image:

This will be possible only if the customer has activated his profile by setting a password to it. Also, prior to that the activation email needs to be sent out from Freshdesk as well. To be sure that this is automated, go to Admin > Workflows> Email Notifications > Requester Notifications > User Activation Email and have it turned on. For the first ticket raised by a contact, the activation email would be sent out automatically using which the contact would be able to set a password for the account.