What is a vertex-transitive polytope? I can not seem to find a proper definition. The one on Wikipedia is overly vague. It just says that roughly, all vertices are the same. What does this mean?
Would a polytope generated as the convex hull of the orbit of some vector in $\mathbb{R}^N$ under some subgroup of the orthogonal group (consider some natural action on $\mathbb{R}^N$) qualify as a vertex-transitive polytope?
In the beginning of this research article (you may want to use a university computer. It is "Symmetry Groups of Vertex Transitive Polytopes" by L. Babai), it is stated that a polytope $P$ is vertex-transitive if the symmetry group of $P$ acts transitively on the vertex set. In other words, if $x,y$ are two vertices, there should exist an automorphism of $P$ such that $g \cdot x = y$. Here the symmetry group of $P$ is defined as the set of orthogonal transformations of $\mathbb R^d$ that takes $P$ to itself.
So the answer to your other question should be yes, almost be definition.