You can pre-populate the users' email addresses and grey-out the field so that they will not be able to edit the email address when the user is logged in. This can be done by greying out the 'Requester' field using a jQuery script.
The code that you'll have to use is -
{% if portal.has_user_signed_in %}
<script type='text/javascript'>
jQuery('#helpdesk_ticket_email').prop('disabled', true);
</script>
{% endif %}
You would have to place this code below the existing code under Admin --> Channels --> Portals --> Customize portal --> Layouts & Pages --> Portal Pages --> New Ticket and then click on Save & Publish.
This option would be available from the Estate plan onwards.