There is no hope of this for any subcanonical topology for which fields are local rings, i.e. rings $A$ such that every covering family of $\operatorname{Spec} A$ contains a split epimorphism. (Fields are local rings for the fpqc topology, so this is true for all coarser topologies, e.g. the Zariski topology.)
Indeed, for any field $K$, we can find a non-trivial subsheaf of $\operatorname{Spec} K$. Let $\phi : K \to L$ be any non-trivial field extension and let $F$ be the sheaf image of $\operatorname{Spec} \phi : \operatorname{Spec} L \to \operatorname{Spec} K$. Then $F$ is a non-trivial subsheaf of $\operatorname{Spec} K$: it is non-empty, since $\phi \in F (L)$, and $\textrm{id}_K \notin F (K)$, because we have $\alpha \in F (A)$ if and only if there exist a cover of $\operatorname{Spec} A$ by affines $\operatorname{Spec} \beta_i : \operatorname{Spec} B_i \to \operatorname{Spec} A$ such that each $\beta_i \circ \alpha : K \to B_i$ factors through $\phi : K \to L$ – but we assumed any cover of $\operatorname{Spec} K$ splits, so $\operatorname{id}_K \in F (K)$ if and only if $\phi : K \to L$ splits, and a field extension splits if and only if it is trivial.
(Note that in the above argument it is not essential that $L$ be a field. It suffices to consider any homomorphism $K \to C$ that is not a split monomorphism, with $C$ non-trivial.)