We show that various flavours of Witt vectors are functorial with respect to multiplicative polynomial laws of finite degree. We then deduce that the $p$-typical Witt vectors are functorial in multiplicative polynomial  \textit{maps} of degree at most $p-1$. This extra functoriality allows us to extend the $p$-typical Witt vectors functor from commutative rings to $Z/2$ Tambara functors, for odd primes $p$. We use these Witt vectors for Tambara functors to describe the components of the dihedral fixed-points of  the real topological Hochschild homology spectrum at odd primes.