I think realistically, such rewards are best seen as binding in honour only.
From a legal perspective, the question would be whether there was offer, acceptance, and performance.
Enforceability might come down to exactly how the reward offer was worded, what you discussed, what you did, and what anyone else might have done.
Unlike Carbolic Smoke Ball (mentioned by @DaleM), which was about a commercial advertisement generating profit for cynical peddlers of quack remedies, this is a case of a victim of a crime acting in a personal capacity to recover stolen property.
The courts might not as readily accept that the reward offer created legally enforceable relations, or that there is any principle benefitted by holding victims of crime to their reward offers in the same way there is a benefit to holding commercial peddlers to their claims.
Personally from what I'm reading, I'm not only suspecting that you might have had some kind of associate connection to the crook who actually stole the vehicle (how many people would pick up a driving licence in a car they're viewing and just pocket it?), but that a $6,000 reward is not quantum meruit for your apparent exertion - it might have been more proportionate and honourable to ask for the owner to pay for a slap-up meal in the diner in exchange for information, rather than the $6,000.
If there's one person you're not going to pay the reward to - come what may, and in the teeth of any coercion or threats - it is the person who stole the vehicle or anyone you think are connected to him (even if only peripherally).
I might be completely wrong, but I mention this logic because it quite possibly explains what the owner might also suspect, and therefore what his attitude will be to ever providing the reward money, and what kind of sympathy a judge may have to a claim.
I'd personally suggest just let the matter be.