All schemes and rings are Noetherian.
Let $ f : X \rightarrow \operatorname{Spec} A $ be proper and $ M $ a finitely generated $ A$-module. Are there any criteria on $ A,X,f $ for there to exist a line bundle $L $ on $ X $ with $ H^0(X,L) \cong M $ as $ A$-modules?
This can fail, for example taking $ A = \mathbb{C} $, $ X = \mathbb{P}^2_{\mathbb{C}} $ and $ M $ a two dimensional vector space.
If you like, you may also assume $ f $ is a resolution of singularities of an affine variety $ \operatorname{Spec} A $ over $ \mathbb{C} $.