There are two very important points you should keep in mind here:
- You are not under any obligation whatsoever to investigate the owner of a vehicle parked on your property.
- You have full rights to tow any unauthorized vehicle off of your property.
So, by far the easiest thing for you to do is to shift all responsibility off of yourself. Make it somebody else's problem.
Try the police first.
The safest thing you can do is simply dial 911 (or try to find a non-emergency number if you live in a major city, but Nebraska suggests just calling 911 directly) and report the abandoned vehicles to police. Their process for declaring a vehicle abandoned can take a bit longer (takes seven days in Nebraska). Essentially they'll document the vehicles' location and tag them, and probably run the license plates (if they come back stolen, they'll be towed by law enforcement immediately). Then they'll come back seven days later and, if the vehicles are still there, have them towed as abandoned vehicles. Law enforcement will sometimes only respond to private parking complaints that are actually on paved surfaces, and it sounds like these vehicles are just parked out in the middle of a field somewhere, so they may not actually care. But it doesn't hurt to check.
If that fails, just have it towed.
If law enforcement says it's ok or doesn't care about the vehicles, the next easiest thing for you to do is to call around to different tow companies, and see if one will tow it off your property for free in hopes of recovering tow costs and other fees from the actual owner of the vehicle, or through sale of the vehicle if it's never claimed. Let them do all the research and contact the owner, or report the vehicle to the police if necessary. You don't need to do any of the work yourself. Sure that doesn't get you any money, but any scenario that gets you money will be a very long process and it sounds like you just want the vehicles gone.
You do not own the vehicles.
The previous owner saying you bought the vehicles with the land is blatantly wrong. Ignore him, completely. By that logic, someone buying an apartment complex would subsequently take ownership of all vehicles on its private parking lot. That's not how vehicle ownership works in any state, and you do not own the vehicles, nor do you have any right to dispose of them. Even if the vehicle is abandoned, there is still a legal process that must be followed to claim ownership of an abandoned vehicle with the state. Unless you really want to take ownership of the vehicle, those processes are probably way more time and effort than you're willing to expend (usually resulting in years of waiting).
Taking it to a scrap yard could be very bad for you.
Since you do not have ownership of the vehicles, you definitely should not take them to a scrap yard. Destroying the vehicles without giving a person the chance to come claim the vehicles could get you into a lot of trouble. You're basically destroying someone else's property. If the person came back looking and found out you destroyed them, they may even be able to press charges against you, the scrap yard, or a combination of both (a Class IV felony in Nebraska, since vehicles are worth more than $1500).
As an aside, any legitimate and reputable scrap yard should outright refuse to destroy the vehicles for you, because you won't be able to provide them with any documents that verify your ownership of the vehicles. Make sure you don't destroy the vehicles in any other way, though.
Again, shift the responsibility.
Don't put yourself into situations if you don't have to. Law enforcement and tow companies deal with this stuff every single day, and are much more qualified to handle this situation in a legal way than you are. Let them take all the responsibility off of you, and don't worry about doing anything yourself. It will make sure you don't do anything illegal, and thus don't open yourself up to repercussions later on down the line.