Received this from an attorney who has been following the CFPB since its conception:

Regarding whether the Bureau can “operate” without a leader, the answer is almost certainly that it can. But it can’t actually “do” anything, including adopting any regulations, without a Senate-confirmed director. This is so because of the Appointments Clause of the Constitution (Article II, Section 2, Clause 2), which empowers the President to appoint certain officials only with the advice and consent of the Senate. An interesting little-known fact about the Bureau is that, unlike nearly every other federal agency, there is no provision in its enabling statute for an under-boss who is also confirmed by the Senate. That means that, if the actual director is disabled, the Bureau will be, too.