No, you may not enter the dunes of Zandvoort. This is to prevent damage. The main function of the dunes is to provide natural protection for the coast against the sea. Without the dunes a large part of the Netherlands would be under water.
In addition, almost all the dunes in the Netherlands is part of a nature reserve. They are an important part of biodiversity. In the dunes live many animals such as rabbits, foxes, deer, lizards and many species of birds. It's official forbidden to pick the plants and to chase animals in the Dutch dunes.
Furthermore you can (of course) walk on the created hiking paths.