Assuming countable choice, normal and conormal bundles are smooth vector bundles

In a slice chart $(x_1,\dots ,x_k,y_1,\dots ,y_c)$ with $S=\{y=0\}$, the induced charts of [L0] make $TM{\mathrm{\mid }}_S$ a smooth vector bundle with local frame ${\mathrm{\partial }}_{x_1},\dots ,{\mathrm{\partial }}_{x_k},{\mathrm{\partial }}_{y_1},\dots ,{\mathrm{\partial }}_{y_c}$, while $TS$ is spanned by the ${\mathrm{\partial }}_{x_i}$. Hence $TS$ is a smooth subbundle and the classes of ${\mathrm{\partial }}_{y_j}$ give a local frame of the quotient $TM{\mathrm{\mid }}_S\mathrm{/}TS$. By [L2], the normal bundle is smooth.

In the same slice chart, the induced cotangent charts of [L0] give the local coframe $d{x}_1,\dots ,d{x}_k,d{y}_1,\dots ,d{y}_c$, and the covectors annihilating $TS$ are exactly the span of $d{y}_1,\dots ,d{y}_c$. These local frames vary smoothly, so the conormal bundle is a smooth subbundle of $T^{\ast }M{\mathrm{\mid }}_S$, hence a smooth vector bundle over $S$.