Falchion had an excellent summation of how the two laws might apply in this case, but I’d like to add one point for context.
In reading laws, there’s a few basic rules for interpreting what laws mean. The first and most important is that we should presume that Congress means what it says and says what it means when writing a law, and also if something is not said that is usually held to be significant as well. So in this case if Congress meant for the CFPB to have a special procedure outside of the Vacancies Act for appointing interim Directors, this issue would be moot if the words “Notwithstanding the Vacancies Reform Act…” had appeared. But they don’t, so we are left with ambiguity.
Following a close second in importance is that if two laws can be read to give full effect to each law, one should attempt to interpret both laws to give full effect to each statute. So in this case, the fundamental question the courts must answer is whether both the Dodd-Frank law and the Vacancies Act can be read in a way such that they do not conflict with each other, while not jumping through extraordinary hoops to do so? (In my opinion, they can, such that Dodd-Frank did not actually do away with the Vacancies Act for this position, which as mentioned before is not in the text of the law, but more of a subjective question of whether there was an implied waiver of the Act.)
It is only if the conflict between two laws is irreconcilable (law A says turn right, law B says turn left) that other tools are used to determine which law should control. Among those tools is that the later in time law would generally (but not always!) be given precedence.
And if you stop to think about this, it makes perfect sense. If we were to adopt the position that every new law should be presumptively read to implicitly wipe out every law that came before it, with no need for the prior laws to be explicitly wiped out; how would you know what the law actually dictates when you pick up a copy of the US Code and start reading it? We would have a mess of arguments as to whether new laws are implicitly wiping out old laws, instead of attempting to read every provision of law to have equal standing (so far as it is possible).
In my opinion, the argument from the CFPB/Warren/Cordray side would be open and shut if Dodd-Frank had said, “The Vacancies Act is waived for this position.” Instead, the main argument seems to be, “Well, that’s what Congress really meant, even if it wasn’t written down.” I think that argument fundamentally fails the first test of, “legislatures mean what they say and say what they mean.”