As long as the old car or other old vehicle was never fitted with seat belts from the factory then you have no obligation legally to install or wear seat belts in the old car or vehicle.
If the old car came with seat belts then you must install new seat belts if there are none and wear them when driving or riding in the car.
Also although the car or other old antique vehicle never came with seat belts you cannot have any kids under 3 riding in the antique car.
If you have kids under 3 or kids that legally are required to ride in a car seat child restraint then you must have at least a seat belt installed to secure the child's car seat child restraint.
But for adults and those who do not legally require to be in a child restraint seat they can ride in the antique car without seat belts.