Operator modules arise naturally from actions of (quantum) groups on operator algebras. With an eye towards approximation properties of (quantum) group actions, we introduce notions of finite presentation which serve as analogues of finite-dimensionality for operator modules over completely contractive Banach algebras. With these notions we then introduce analogues of the local lifting property, nuclearity, and semi-discreteness. For a large class of operator modules, we show that the local lifting property is equivalent to flatness, generalizing the operator space result of Kye and Ruan. We pursue applications to abstract harmonic analysis, where, for a locally compact quantum group $\mathbb{G}$, we show that $A(\mathbb{G})$-nuclearity of the inclusion $C^*_{\lambda}(\mathbb{G})\rightarrow C^*_{\lambda}(\mathbb{G})^{**}$ and $A(\mathbb{G})$-semi-discreteness of $VN(\mathbb{G})$ are both equivalent to co-amenability of $\mathbb{G}$. In the co-commutative setting, when $A(\mathbb{G})=A(G)$ is the Fourier algebra of a locally compact group $G$, we establish the equivalence between $A(G)$-injectivity of the crossed product $G\bar{\ltimes}M$, $A(G)$-semi-discreteness of $G\bar{\ltimes} M$, and amenability of $W^*$-dynamical systems $(M,G,\alpha)$ with $M$ injective.