If you look at a family of elliptic curves like $y^2 = x^3 - x-t^n$ as a surface $X$ mapping to a curve $C$ with parameter $t$, and $Y$ the fiber over $0$, then the neighborhood modulo $\mathcal I_Y^n = (t)^n= (t^n)$ is trivial, hence equal to the $n$'th formal neighborhood of $Y$ in the total space of $N_{Y/X}$, but the neighborhood modulo $\mathcal I_Y^{n+1}$ is not. This shows that no finite list of obstruction classes will describe the formal neighborhood.

If the $n$th' order formal neighborhood is isomorphic to the $n$th order formal neighborhood in the normal bundle, then the ring of functions on it is isomorphic to $$\mathcal O_Y \oplus N_{Y/X}^\vee \oplus \operatorname{Sym}^{2} N_{Y/X}^\vee \oplus \dots \oplus \operatorname{Sym}^{n-1} N_{Y/X}^\vee $$ and the $n+1$st order neighborhood is locally the sum of that with $\operatorname{Sym}^n N_{Y/X}^\vee$ and then automorphisms respecting the algebra structure and the filtration but not the direct sum structure are given by $\operatorname{Sym}^n N_{Y/X}^\vee$-values derivations on that sum. A derivation is determined by the Leibnitz rule and its value on the first two factors, so the local automorphisms are given by $$\mathcal T_Y \otimes \operatorname{Sym}^n N_{Y/X}^\vee + N_{Y/X} \otimes \operatorname{Sym}^n N_{Y/X}^\vee$$ and so you obstruction classes in $H^1(Y, \mathcal T_Y \otimes \operatorname{Sym}^n N_{Y/X}^\vee )$ and $H^1(Y, N_{Y/X} \otimes \operatorname{Sym}^n N_{Y/X}^\vee)$, I think.

The first obstruction controls when the formal neighborhood can be viewed as a fibration over $Y$ and the second obstruction controls when you can identify the fibers with the normal bundle.