In the category of schemes, the equalizer of two morphisms $f,g : X \to Y$ is always a locally closed immersion into $X$ (since this is just $X \times_{Y \times Y} Y$ and $\Delta : Y \to Y \times Y$ is a locally closed immersion). What about the converse, is every locally closed immersion some equalizer?

Certaily every open immersion $U \to X$ is an equalizer, namely of the two maps $X \to X \cup_U X$. For closed immersions $Z \to X$, a similar construction works: The pushout $X \cup_Z X$ exists also in this case, see here, and is constructed locally, so that it is enough to remark that $A \to A/I$ is the coequalizer of the two maps $A \times_{A/I} A \to A$ in the category of rings. I general we have a composition of a closed immersion followed by an open immersion, but I'm not sure if we can do the same construction.