I was able to figure out my problem.
The CRON errors in /var/log/syslog were not being saved because I did not have a mail-transfer agent installed. I installed postfix which allowed the error output to be sent to /var/mail/root.
The problem was that I had set up flexget to use the config file in my user folder. Root's crontab was running flexget, and flexget was looking for the config file in /root/.flexget/ which didn't exist. I got the following error output in /var/mail/root
So I changed root's crontab to specify the config file like so:
2012-09-02 15:00 INFO manager Tried to read from: /usr/local, /root/.flexget, /root/.config/flexget
2012-09-02 15:00 CRITICAL main Failed to find configuration file config.yml
That fixed my issue. Hopefully this solution can help someone else in the future.
0 */5 * * * /usr/local/bin/flexget -c /home/myusername/.flexget/config.yml --cron