A vector field along an embedded submanifold extends to a neighbourhood and globally when the submanifold is closed

By [L1], every point of $S$ has a slice chart $(U_{\alpha },x_{\alpha })$ in which $S\cap U_{\alpha }$ is given by $x_{\alpha }^{k+1}=\cdots =x_{\alpha }^n=0$. Shrink the chart around the point to a product coordinate box, whose tangential projection remains in the slice. On that slice, $Y$ has smooth coordinate components, so extending those coefficient functions constantly in the normal coordinates defines a smooth vector field ${\tilde{Y}}_{\alpha }$ on the smaller chart.

The open sets $U_{\alpha }$ cover $S$. Choose a smaller open neighbourhood $U\subseteq {\bigcup }_{\alpha }{U}_{\alpha }$ of $S$, and by [L2] choose a partition of unity $({\rho }_{\alpha })$ on $U$ subordinate to $(U_{\alpha }\cap U)$. Then $\tilde{Y}:={\sum }_{\alpha }{\rho }_{\alpha }{\tilde{Y}}_{\alpha }$ is a smooth vector field on $U$, and on $S$ the local fields all equal $Y$, so $\tilde{Y}{\mathrm{\mid }}_S=Y$. Countable choice is used by the construction of [L2], including selection of the countable bump family.

Assume now that $S$ is closed. Step 2.1 supplies a smooth extension $\tilde{Y}$ on an open neighbourhood $U$ of $S$.

Because $S$ is closed in $M$ and lies in the open set $U$, [L3] gives a smooth function $\chi :M\to \mathbb{R}$ with $\chi =1$ on $S$ and $\mathrm{supp}\chi \subseteq U$. Define $\widehat{Y}:=\chi \tilde{Y}$ on $U$ and $\widehat{Y}:=0$ on $M\setminus \mathrm{supp}\chi$. This is a smooth global vector field and restricts to $Y$ on $S$.

Therefore every smooth vector field along an embedded submanifold extends to a neighbourhood, and to all of $M$ when the submanifold is closed.