A group of students from a Torah Acaqdemy, adhering to their Jewish
faith, does not compete in sports or academic events on Saturday, the
sabbath.  Its mock trial team competes throughout the year on Tuesdays
and Wednesdays, and wins the state mock trial championship, giving it
the right to compete in the National competition, the final rounds of
which are held on Saturday.

Is there an exception in Jewish law that would permit them to compete
on the sabbath?

There is no exception in Jewish law for such a "trivial" (no offense
intended) reason as yours.

The first obvious thing to do is ask the sponsor to change the day of
the finals. If it will not, tell the sponsor you are going to
immediately retain counsel, that this will not be good publicity for
the organization, and they should rethink their position. If they are
still recalcitrant, get a really good lawyer and go after the fools.