You appear to not be using the templates and overrides on your site so this makes debugging much more difficult ...
Have you ever been able to checkout?
What version of Zen Cart are you running?
Clean install or an upgrade? If an upgrade, from what version and how did you do it?
If you have not changed any core code and only changed files in the /classic directories ... you can try a test to see if this is your changes or a bad, corrupt, missing file or something wrong with your email ...
Carefully rename the directory:
/includes/templates/classic
to the name:
/includes/templates/classic_HIDE
While the site will look very strange, can you check out now?
NOTE: Rename it back to /classic to go back to your current template ...