Impatient for an answer huh.
You script doesn't work because it doesn't make sense. < and << are for I/O redirection. You're not redirecting anything.
for file in $(ls -lt /tmp);do
echo $file | mail firstname.lastname@example.org
That will get you a bunch of emails each containing the name of one file that's in /tmp. If you want one email that contains the output of the ls command then you're really over complicating things as all you need is
ls -lt /tmp | mail email@example.com