Under the 14th Amendment, the Bill of Rights applies to state law. If there were any BoR problems with car registration by states, it would be just as relevant as BoR problems with national gun registration.
The question is, are there any Constitutional restrictions on a federal gun registry? That's a pretty high standard. If the standards were that laws cound not be a bad idea and not posess a high potential for abuse, very few federal laws would pass muster by SCOTUS. We here at FR tend to be constitutional literalists. Supreme Court judges are more akin to Humpty Dumpty - words mean what they want them to mean, so if we want to prove that a national gun registry is unconstitutional, we have to point to specific SCOTUS rulings regarding the potential for abuse, rather than talk in concrete BoR statements. It sucks, but that's how things work nowadays.
Well let's reason this out.
The 4th Amendament:
"The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no warrants shall issue, but upon probable cause, supported by oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized."
The 10th Amendment:
"The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people."
A better question would be "what gives the federal government the right to regulate any 'affects' of the people?"