The most likely culprit is a corrupt, missing or damaged e-mail template for the postcard contents. In the templates directory for your chosen theme (or the templates_default directory if your not using a custom theme) you should locate a template file with the path and name:
email.templates/postcard.content.tpl
This is where the code for the postcard e-mail contents is stored and if this file is damaged or missing, any postcards you send from your website will be completely empty. To fix the problem, re-upload all the template files again using FTP.