There was a 7-2 decision in that case that suggests otherwise. Thomas' concurring opinion was the only one that directly addressed the constitutionality of the Vacancies Act, so it wasn't a critical element of the overall decision in the NLRB case.
Federal regulations are laws in all but name. The walk like laws. They take like laws. They smell like laws. They taste like laws. They can impose fines like laws. And they can imprison people, just like laws.
The Constitution grants absolutely zero lawmaking authority to the Executive Branch. That authority is granted solely to the Congress. Not to the IRS. Not to the FTC. Not to the FDA. Etc.
A great example of a law that would fall, unless Congress can Constitutionally delegate it's authority to the Executive Branch, is the War Powers Act. There are so many others. More: From Administrative State to Constitutional Government (Heritage Foundation)