= Solution
Choose local coordinates $y^1,\ldots,y^{D-1}$ on the <hypersurface> $\Sigma$, and let $\Phi_s$ be the <local flow> of the <vector field> $X$. The map $F(s,y)=\Phi_s(p(y))$ follows the <integral curves of a vector field> starting at $p(y)\in\Sigma$. At $s=0$, its differential maps the $s$ direction to $X$ and the remaining directions to a basis of $T_p\Sigma$. Transversality makes these vectors linearly independent, so the <inverse function theorem> makes $F$ a local <manifold chart>.
Set $x^0=s$ and transport the $y^i$ unchanged along each <integral curve of a vector field>. Then \b[the adapted coordinates satisfy]
$$
\boxed{X=\frac{\partial}{\partial x^0},\qquad X(x^i)=0.}
$$
The <flow-box theorem> gives this construction locally. The parameter is the flow parameter, and need not be <arc length>; a global <manifold chart> would require additional hypotheses.
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