There were VIN numbers or equivalents of various types in all autos long before 1981. Motors will have seriel numbers, and so will the bodies of vehicles. Mechanics who have experience on the specific type of vehicle will know where to check. A "dune buggy" is not a standard model, of course, most will have the rear end of a VW van instead of a "bug", but with a "bug" body. The van rear ends have sturdier axles. You'll want the numbers off the motor and the body, but the motor could easily be a replacement, this is common with Volkswagens.
If you have the original title and registration there will be (or should be) a list of all the owners of the vehicle. Each new owner's name must be added when the car is sold. If the title is a replacement title, as if the original were lost at some time, then you may not have a list of all the previous owners. This means you're only recourse is the "detective method"- you ask the person you bought it from where he/she got it, go to that person and find out who they bought it from, etc. Tracking down the owners may not be feasible or possible, but you could get lucky. Unless in your tracking you find Paul Newman or unless you find some previous owner with actual provable documentation that Paul Newman owned the vehicle at some point, you will have no proof. I hope you didn't pay extra for this car just because the seller told you it was once owned by Newman. If it really means a lot to you to find out hiring a private investigator may seem worthwhile to you, but it seems like a lot of expense and trouble to go to.