Let $\text{PreSh}(X, \mathsf{A})$ and $\text{Sh}(X, \mathsf{A})$ be the categories of $\mathsf{A}$-valued presheaves and sheaves on $X$, respectively. Here, $\mathsf{A}$ is a category such that we can perform sheafification, and $X$ is a topological space (or a site if you prefer, though I don't know much about these).
The subcategory $\text{Sh}(X, \mathsf{A}) \hookrightarrow \text{PreSh}(X, \mathsf{A})$ is reflective by assumption, i.e., the embedding has sheafification as a left adjoint (called a reflector in general). Recall that a reflective subcategory is said to be exact if in addition the reflector is left exact (and hence exact).
If $\mathsf{A}$ is the category $\mathsf{Set}$, then $\text{Sh}(X, \mathsf{A})$ is an exact reflective subcategory (i.e. sheafification is exact). The embedding functor $\text{Sh}(X, \mathsf{A}) \hookrightarrow \text{PreSh}(X, \mathsf{A})$ is left exact by assumption, but I am not sure if it is right exact as well.
Now what if $\mathsf{A}$ is some other category, such as $R$-$\mathsf{Mod}$? When can we say that the sheafification and the embedding functors $\text{Sh}(X, \mathsf{A}) \hookrightarrow \text{PreSh}(X, \mathsf{A})$ are exact? How about if we are dealing with a ringed space $(X, \mathcal{O})$ and sheaves of $\mathcal{O}$-modules?
Sorry if this is a silly question; I'm studying differential geometry so I am not very familiar with sheaves yet.