The answer is yes.

The fact that any polytope is affinely equivalent to a section of a simplex is well-known (see the answer by Tobias Fritz).

Now in any simplex with vertices $(v_i)$ we may consider projective transformations via reweighting barycentric coordinates: given positive numbers $(a_i)$, such a transformation is given by $$ \sum \lambda_i v_i \mapsto \frac{\sum a_i \lambda_i v_i}{\sum a_i \lambda_i}.$$

Projective transformations act transitively on the interior of the simplex, so it follows that any polytope is combinatorially equivalent to a section of some regular simplex through it center.

What I don't know is whether every polytope is affinely equivalent to a section of some regular simplex through it center.

The answer is yes.

The fact that any polytope is affinely equivalent to a section of a simplex is well-known (see the answer by Tobias Fritz).

Now in any simplex with vertices $(v_i)$ we may consider projective transformations via reweighting barycentric coordinates: given positive numbers $(a_i)$, such a transformation is given by $$ \sum \lambda_i v_i \mapsto \frac{\sum a_i \lambda_i v_i}{\sum a_i \lambda_i}.$$

Projective transformations act transitively on the interior of the simplex, so it follows that any polytope is combinatorially equivalent to a section of some regular simplex through its center.

What I don't know is whether every polytope is affinely equivalent to a section of some regular simplex through its center.