Welcome to AC.
Do you have access to a computer you can use to log in to the webmail access for the email address? We need to see if the emails still reside on the server. If they do, you can then just redownload them once your PC is functioning again.
Can you export them from the mail client on your phone? That may depend on what mail client it is but it's not really likely. What is more likely is that the mail client is not set to delete the emails from the server once it's accessed them so they may still be there. I suppose it's possible the phone still has a copy of the emails but the PC deleted them when IT downloaded them. That would pose a challenge.