Even in the absence of an electronic TMF I would advise looking at solutions that enable you to retain the emails digitally. You need to ensure you are meeting the relevant regulatory guidance for managing documents electronically but these are not particularly arduous. An option might be to manage emails within your email system while the trial is ongoing – BUT ensuring you have appropriate processes to ensure they are actually managed and not just dumped into folders! – and then exported at the end of the trial and saved to a suitable repository e.g. read-only, access-controlled and designated area of a network storage device.
A critical benefit that you achieve by retaining digitally is the ability to search through the content. This could be invaluable in the event of an inspection.
Whatever you come up with as a solution, make sure it is properly documented and any tech qualified for use.