= Solution
The free embedding fields have
$$
\partial X^\mu(z)\partial X^\nu(w)
\sim-\frac{\alpha'}2\frac{\eta^{\mu\nu}}{(z-w)^2}.
$$
In $T_X(z)T_X(w)$, the double contraction of the two derivatives in each normal-ordered stress tensor has two pairings. It gives
$$
T_X(z)T_X(w)\sim\frac{D/2}{(z-w)^4}+\cdots.
$$
The <stress-tensor operator-product expansion> therefore has central charge $c=D$. Contour integration against the infinitesimal holomorphic vector field gives
$$
\delta_vT_X=v\partial T_X+2(\partial v)T_X+\frac{D}{12}\partial^3v,
$$
so in the notation of the question
$$
\boxed{c_X=\frac D{12}}.
$$
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