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Let $H$ denote a finite index subgroup of the modular group $\Gamma$ and let $\rho$ denote a finite-dimensional complex representation of $H.$ Let $M\left(\rho \right)$ denote the collection of holomorphic vector-valued modular forms for $\rho$ and let $M\left(H\right)$ denote the collection of modular forms on $H$. Then $M\left(\rho \right)$ is a $\mathbb{Z}$-graded $M\left(H\right)$-module. It has been proven that $M\left(\rho \right)$ may not be projective as a $M\left(H\right)$-module. We prove that $M\left(\rho \right)$ is Cohen-Macaulay as a $M\left(H\right)$-module. We also explain how to apply this result to prove that if $M\left(H\right)$ is a polynomial ring, then $M\left(\rho \right)$ is a free...