It seems the 'tap tempo' function is only offered in the Pro version of Logic. Seems odd because it's rather easy to find a free 'tap tempo' app online. Here's one example that seems to do the trick.
http://www.grantedsw.com/midiclock/
I'd love to see Express users offered this small bit of convenience as a built-in feature in a future update. I won't hold my breath or anything though.