Is it possible to play league games with the base Ninja-all-stars game?
So having read the rules, I've got my game not so long ago, I'm able to answer.

The base game provides 4 teams that correspond to a shrine, which is roughly - but not exactly - the same as playing the Clan attuned to that specific element.

Without the clan box, it's impossible to play Onis (no miniatures compare in the base box), Heroes and most ronins. You can proxy (and in fact is the solution given by the rules) any clan with the base set - except Onis, Heroes and any ronin which happened to not be a Samurai.

So it's doable to play league games with only the base game, but is kinda hard to read, unless people limitate themselves, and not all is available even then.