We know that the evaluation functor $\def\mod{\textrm{-mod}} \Gamma(U, -):Qcoh(X) \to \mathcal{O}_X(U)\mod$ is a left exact functor preserving limits. We also know that the category of all quasi-coherent sheaves on scheme is locally finitely presented. So we may use "Adjoint Functor Theorem" and deduce that there is a left adjoint for the evaluation functor.
Is there any explicit description for this adjoint?
The answer is true if we replace $Qcoh(X)$ by the category of sheaves on $X$ (see section 2 of "Relative Homological Algebra in Categories of Representations of Infinite Quivers" by S. Estrada)