Is there a notion of "schemeification" analogous to that of sheafification of a presheaf?

There is an "affine schemification", a left adjoint to the inclusion of affine schemes into locally ringed spaces. Indeed, given a locally ringed space $X$, the left adjoint just sends $X$ to $\operatorname{Spec} \mathcal{O}_X(X)$. To sketch the proof, there is a canonical map $X\to\operatorname{Spec}\mathcal{O}_X(X)$, since each point $p\in X$ can be sent to the prime ideal of global functions which are in the maximal ideal of $\mathcal{O}_{X,p}$. You can then verify that this actually naturally can be made into a morphism of locally ringed spaces and is initial among all morphisms from $X$ to affine schemes. See Theorem V.3.5 of Peter Johnstone's Stone spaces for more details.

However, there is not a "schemeification" if you allow all schemes and not just affine schemes. The problem is that schemes do not have arbitrary limits. For instance, an infinite product of non-affine schemes typically does not exist; see this MO question. On the other hand, the category of locally ringed spaces has all limits (this is not obvious; see Corollary 5 of this paper). It follows that there can be no schemeification functor, since a reflective subcategory must be closed under limits. Explicitly, if you have a diagram of schemes that has no limit in schemes, then its limit in locally ringed spaces has no schemeification, since its schemeification would be a limit in schemes.

For some related discussion, see this very nice answer by Martin Brandenburg.