Does someone know if the following is true, and if so could you provide a reference.
Let $X$ be a smooth $S$-scheme, let $Z$ be a closed subscheme of $S$, and let $t : Z \to X$ be an $S$-morphism. Then, there exists a closed suscheme $Z \subset Y \subset X$ such that $Y \times_S Z = Z$ and $Y \subset X \to S$ is étale on neighbourhood of $Z$.
I tried to prove it but did not understand how to construct $Y$. Is there a discussion of such question in either EGA or on the Stacks Project?
As far as I can see this isn't yet in the Stacks project. Also, the statement as you have it is probably too strong; in general I would only expect the existence of a $Y$ Zariski locally on $X$. Anyway, the closest thing I could find is the material in Section 055S -- you would need to change the arguments a bit in order to make it work for your question.