A morphism of sheaves $f: X \to Y$ in the fpqc topology on $Aff$ [covers are finite universally epimorphic families $(Spec(R_i)\to Spec(R))_i$ in $Aff$ with each morphism $Spec(R_i)\to Spec(R)$ flat] is representable by open immersions of schemes if and only if:
1) for all local schemes $Spec(R)$ with closed point $Spec(k)$ (in the category $Sh/Y$) the natural map $Hom_Y(Spec(R), X) \to Hom_Y(Spec(k), X)$ is bijection [or with condition 3) below, just surjective].
2) it is locally finitely presented (in the presheaf theoretical sense).
3) it is a monomorphism.
Notes: a) Conditions 1) and 2) only are equivalent to the map being representable by "local isomorphisms of schemes" for example the map $X \to \mathbf{A}^1$ where $X$ is the affine line with the original double and the map just folds in the double point. However, these maps are 'no good' (i.e. they do not satisfy fpqc descent).
b) A scheme is a sheaf $X$ in the fpqc topology on $Aff$ such that there exists a cover (in the canonical topology on $Sh$) by affine schemes $(Spec(R_i)\to X)_i$ with each map $Spec(R_i)\to X$ satisfing the conditions above.
c) I haven't checked but I'm pretty sure this will work with the other natural topologies (fppf, etale, Zariski).
