A moduli problem inspired by Stein factorization I'm not sure a moduli space would be the right thing to expect. However, it should be easy to exactly characterize what sheaves these are. Up to isomorphism, these are exactly the sheaves of the form $O_X(D)$ where $D$ is effective and exceptional. For instance if $Y$ is $\mathbb{A}^2$ and $X$ is the blowup of $Y$ at the origin, then the set of such sheaves up to isomorphism is in bijection with the natural numbers.

reference for weighted blow-up In this generality, I'm not sure that much can be said besides that you are blowing up a subscheme. What examples have you already tried?