Can I set form submissions to automatically delete after download?

It's not possible to delete submissions as soon as you download them, but you may alternatively send them to a third party could service such as Google Drive or Dropbox and auto delete the submissions as soon as they are received. 

Please find the auto-delete app here: https://apps.jotform.com/app/auto_delete_submissions 

And this link will provide you with our current third party cloud integrations: https://apps.jotform.com/category/file_sharing_and_storage 

I hope this helps.