I think the answer is 'No'.

A very simple counterexamples is as follows:

Let $M = \mathbb R^2$. Let $X$ be the stratification of $M$ given by two strata, origin (0,0) and $\mathbb R^2 \setminus (0,0)$.

Now consider the spiral $\mathcal A$ in $\mathbb R^2$ defined by the condition that the tangent to the spiral makes a constant angle with the radial vector. In polar coordinates the equation of such a spiral can be given by $r - b\theta =$ constant. Take $W$ to be this spiral $\mathcal A$ union the origin and notice that $S_W = \{\mathcal A, (0,0)\}$ is not a Whitney stratification.

I think the answer is 'No'.

A very simple counterexample is as follows:

Let $M = \mathbb R^2$. Let $X$ be the stratification of $M$ given by two strata, the origin $(0,0)$ and $\mathbb R^2 \setminus (0,0)$.

Now consider the spiral $\mathcal A$ in $\mathbb R^2$ defined by the condition that the tangent to the spiral makes a constant angle with the radial vector. In polar coordinates the equation of such a spiral can be given by $r - b\theta =$ constant. Take $W$ to be this spiral $\mathcal A$ union the origin and notice that $S_W = \{\mathcal A, (0,0)\}$ is not a Whitney stratification.