Possible Solution!
I might have found a way to make this work, but it takes quite a bit of work for something so simple. I had to go into the API settings, add a key and passphrase for the local IP address (127.0.0.1), then go into the crontab and instead of using:
php5 /var/www/support/api/cron.php
You have to use this instead:
wget -q -O /dev/null --user-agent=527C27522FE311B195E2F83F47FADB8A http://localhost/support/api/cron.php
I stumbled upon this by using running php5-cgi /path/to/cron.php and noticing that it gave me a 500 Internal Server Error. Then I tried it with wget, where I got a 403 Forbidden. So I went through the code and noticed that in order to pass an API key, you have to rewrite the user agent string with the key. I figured it wouldn't hurt anything to try, and that did the trick.
I don't know why you need to supply an API key to run the cron.php file on the local machine, but that did the trick. I smell a bug!!!!! I've seen a couple other unanswered posts having the same issue as me, so I know it's not just me.
So the final result in the crontab (sudo crontab -e) would be the following:
*/5 * * * * wget -q -O /dev/null --user-agent=527C27522FE311B195E2F83F47FADB8A http://localhost/support/api/cron.php
But you have to replace the the API key above with the key that gets generated when create the API key for your localhost (127.0.0.1).
Hope this helps others experiencing the same issue.