Let T be a bounded multilinear operator on some product of Lq(ℝn) spaces. Assume that T has a non-smooth associated kernel which satisfies certain weak regularity conditions but not regular enough to fall under the scope of the standard multilinear Calderón–Zygmund theory. The main aim of this paper is to establish a sufficient condition on the kernel of T so that the commutator of a vector BMO function (data missing) and T is bounded on certain product Lq(ℝn) spaces. We obtain boundedness of the commutator of (data missing) and T by first proving certain pointwise estimates on the Fefferman–Stein sharp maximal operator. An important example of multilinear operators which satisfy our kernel conditions is the maximal mth order Calderón commutator.