The logistic vector random field is introduced in this paper as a scale mixture of Gaussian vector random fields, and is thus a particular elliptically contoured (spherically invariant) vector random field. Such a logistic vector random field is characterized by its mean function and covariance matrix function just as in the case of Gaussian vector random fields, and so it is flexible enough to allow for any possible mean structure or covariance matrix structure. We also derive covariance matrix functions whose direct and cross covariances are of the logistic type.