I was able to get it working.
I finally found the answer here:
http://osticket.com/forums/showpost.php?p=37198&postcount=5(http://osticket.com/forums/showpost.php?p=37198&postcount=5)
what I needed to do was to change the wget to use POST, like this:
*/1 * * * * wget -q -O /dev/null --post-data '' --header='X-API-KEY:<my API key>' http://<domain and path>/api/tasks/cron
that worked for me. I'll probably change the interval from 1 minute to something higher, but having it low makes testing go faster.