Yesterday I typed up a big post on thread #26 since I saw some people seemed to suggest it's simple to look up a car using a "national database". I accidentally lost the post and didn't feel like typing it up again, but here goes...
From my experience having worked in criminal and federal courts, a combination of local and national databases may be used based on what facts LE already have or don't have. The "national database" is not as streamlined or complex as one might think.
Their job to identify the owner of a car would be fairly simple if they have the license plate number by using the relevant state DMV or police records. If LE knows the license plate number of the Hyundai, then they certainly already know who owns the car. From there, knowing the VIN number also makes it easier to track down the car later should someone ever try and register.
It would be more complicated if they only have the make/model/color/estimated year. In that case, you're still mostly going to need to rely on local data but you probably don't know where the car was registered without the license plate. The National Motor Vehicle Information System maintained by the Federal gov't doesn't always have specific details about the make of cars. In that case, you need to know where the car was registered to look at more detailed state or local records. Say LE has a list of every white Hyundai owner's name, address, and license plate number for all of Idaho and the surrounding states, well, that may not tell you anything useful if the car was stolen, the licenses plates were switched, the car was from another state, etc.
The Federal gov't only maintains two databases related to cars, one for titling info (which not every state even reports to yet) and another for stolen cars and revoked/suspended drivers. More detailed info than that will only be kept locally in various ways depending on the state. If the FBI has a secret national database of car info then it's probably because they bought the data from car insurance companies or scrapped it with surveillance. The ACLU would certainly want to have a word.
Each state does things differently as well, there may be more ways car information is documented publicly. In my state, we pay personal property on our cars. I can go to my county website right now, type in a name or address, and see a basic description of the car that person pays property taxes on. Information held by the department of revenue on cars in my state doesn't need a warrant to access, it can simply be ordered. Information held by the assessor's office that can be found with the basic search function online obviously doesn't need a warrant to access, but it can get tricky, if LE needs to run a more specific "back end" type of search on the assessor's data then they may need a digital warrant. Again, this is for my state/county, not Idaho.
A cursory look through of the Latah County website it does not appear that they pay personal property taxes on cars in that county and likely don't have that additional documentation on cars. This is another reason knowing where a car was registered is important.
TLDR; there is not a super cohesive "national database" of cars. Having something physical to track and spot is more important than any information they could glean about the car through a database imo.