Let $X$ be a smooth variety over a field $\Bbbk$, and let $Y, Z \subset X$ be closed reduced subschemes of the same dimension, both of which are local complete intersections. Is $Y \cup Z$ necessarily a local complete intersection?
The union of two planes in $\mathbb A^4$ which meet at a point is not Cohen--Macaulay, and so in particular not a local complete intersection.
More generally, any smooth subvariety of a smooth variety is a local complete intersection, so any non-Cohen--Macaulay subvariety whose components are smooth gives an example of a union of local complete intersections which is not itself a local complete intersection.
[Added: Your question has the caveat "with an appropriate scheme structure"; is there any other structure besides the reduced scheme structure on the union which you had in mind?]
No. Let $X=\mathbb A^3_k$ with system of coordinates $t_1, t_2, t_3$, let $Y$ be the hyperplan $t_3=0$, $Z$ be the union of the $t_1$-axe and the $t_3$-axe. Then $Y$ and $Z$ are complete intersection, but $Y\cup Z$, whenever the structure you endowe with, can not be a local complete intersection at the origin because its irreducible components don't have the same dimension.