Let ψ be an analytic self-map of the complex unit disk and X a Banach space.This paper studies the action of composition operator Cψ: f → fo ψ on the vector-valued Nevanlinna classes N(X) and Na(X). Certain criteria for such operators to be weakly compact are given. As a consequence, this paper shows that the composition operator Cψ is weakly compact on N(X) and Na(X) if and only if it is weakly compact on the vector-valued Hardy space H1 (X) and Bergman space B1 (X) respectively.