Extensions of quasicoherent sheaves are quasicoherent.

Harts theorem 5.7: Given an exact sequence $0 \to \mathscr F_1 \to \mathscr F_2 \to \mathscr F_3 \to 0$ of sheaves on $X = \mathrm{spec} A$, if $\mathscr F_1$ and $\mathscr F_3$ are quasicoherent, then so is $\mathscr F_2$.

We get an exact sequence $0 \to M_1 \to M_2 \to M_3 \to 0$ (where each is the module of global sections of each $\mathscr F$, and then we get the diagram

$$\require{AMScd} \begin{CD} 0 @>>> \tilde M_1 @>>> \tilde M_2 @>>> \tilde M_3 @>>> 0 \\ @. @VVV @VVV @VVV @. \\ 0 @>>> \mathscr F_1 @>>> \mathscr F_2 @>>> \mathscr F_3 @>>> 0 \\ \end{CD}$$

Where does the middle morphism come from? The other two are isomorphisms from the equivalence of quasicoherent sheaves and $A$-modules, but where does the middle one come from?

For $X = \text{Spec}(A)$ the functor $\Gamma(X,-): {\mathscr O}_X\text{-Mod}\to A\text{-Mod}$ is right adjoint to the sheafification functor $\widetilde{(-)}: A\text{-Mod}\to {\mathscr O}_X\text{-Mod}$. In particular, for any ${\mathscr O}_X$-module ${\mathscr F}$ with global sections $M$ you have a canonical and natural counit morphism $\widetilde{M}\to{\mathscr F}$. This gives the transformations in (as well as the commutativity of) your diagram. To see the exactness of the upper row, you need to apply Proposition 5.6 in Hartshorne.
Explicitly, suppose $M$ is an $A$-module, ${\mathscr F}$ is an ${\mathscr O}_X$-module and $\varphi: M\to {\mathscr F}(X)$ is given. Then, by definition, given any $f\in A$ the sections ${\mathscr F}(D_f)$ are a module over ${\mathscr O}_X(D_f) = A_f$, hence $M\to {\mathscr F}(X)\to{\mathscr F}(D_f)$ extends canonically to a morphism $M_f\to {\mathscr F}(D_f)$ of $A_f$-modules. As $\widetilde{M}(D_f)=M_f$ by definition, these constitute the desired extension $\widetilde{M}\to{\mathscr F}$.