While the answer to the question of OP is "no", I would like to mention a question raised by Grünbaum and Shephard in "Some problems on polyhedra.", J. Geom. 29 (1987), no. 2, 182–190, for the information of those who may be interested.

They say that a polytope $P$ is of DV-type if it has a combinatorially equivalent realization such that every vertex $v$ of $P$ is an interior point of the face of a dual $P^*$ that corresponds to $v$ in the duality. The questions

1. Is every polyhedron of DV-type?
2. Does every polyhedorn of DV-type have a dual of DV-type?
3. Which polyhedra of DV-type are also of SV-type?

I'm not sure about the current status of the question.

While the answer to the question of OP is "no", I would like to mention a question raised by Grünbaum and Shephard in "Some problems on polyhedra.", J. Geom. 29 (1987), no. 2, 182–190, for the information of those who may be interested.

They say that a polytope $P$ is of DV-type if it has a combinatorially equivalent realization such that every vertex $v$ of $P$ is an interior point of the face of a dual $P^*$ that corresponds to $v$ in the duality. The questions

1. Is every polyhedron of DV-type?
2. Does every polyhedorn of DV-type have a dual of DV-type?
3. Which polyhedra of DV-type are also of SV-type? (inscribable w.r.t. a sphere)

I'm not sure about the current status of the question.