No, peanuts do not grow on a vine.
Peanuts are unusual in that they are legumes, but unlike beans or peas that grow above ground, peanuts develop underground. The peanut plant itself grows above ground, but after pollination, the flower's ovary elongates and forms a "peg" that pushes into the soil. The peanut pods then develop and mature underground. So, while the plant grows above ground, the peanuts themselves do not grow on a vine; they grow beneath the surface.
