A complex contact structure gamma is defined by a system of holomorphic local 1-forms satisfying the completely non-integrability condition. The contact structure induces a subbundle Kerry of the tangent bundle and a line bundle L. In this paper, we prove that the sheaf of holomorphic k-vectors on a complex contact manifold splits into the sum of O(boolean AND(k) Ker gamma) and O(L circle times boolean AND(k-1) Kerry) as sheaves of C-module. The theorem induces the short exact sequence of cohomology of holomorphic k-vectors, and we obtain vanishing theorems for the cohomology of O(boolean AND(k) Ker gamma).