I think the answer is **no** for $d\ge 4$.

Looking things backward, you start from a regular simplex $\Delta$ in $\mathbb{R}^{d-1}\times\{0\}$ and you choose a height for each of its vertex, defining a $(d-1)$ simplex in $\mathbb{R}^d$ as the convex hull of the points at the given height above each vertex of $\Delta$; you want to realize every possible $(d-1)$ simplex geometry that way.

On the one hand, you only have $d$ heights to choose, and with the vertical translation invariance this leaves you with $d-1$ degrees of freedom. On the other hand, the moduli space of $(d-1)$ simplices (up to similarity), if I am not mistaken (*edit: I was, now corrected*), has dimension $d(d-1)/2-1$. For $d=3$ you get an equality, but as soon as $d>3$ there are too many simplices for them to be projected on only one shadow.