No, a fixed number of added vertices can change the John ellipsoid by an arbitrary amount when added to arbitrarily many already known vertices: consider a very thin regular n-gon prism centered about the origin. Now we can add two new vertices in a way that doesn't change the polyhedron much (near existing vertices, say), or we can add two new vertices along the axis of the prism very far away from it (in a symmetric way, so that the polyhedron is still centered about the origin). The two possibilities have very different John ellipsoids, so you cannot approximate the John ellipsoid before knowing where these vertices are.

No, a fixed number of added vertices can change the John ellipsoid by an arbitrary amount when added to arbitrarily many already known vertices: consider a very thin regular n-gon prism centered about the origin. Now we can add two new vertices in a way that doesn't change the polyhedron much (near existing vertices, say), or we can add two new vertices along the axis of the prism very far away from it (in a symmetric way, so that the polyhedron is still centered about the origin). The two possibilities have very different John ellipsoids, so you cannot approximate the John ellipsoid before knowing where these vertices are.

If you are sampling the vertices uniformly at random, then you have a good chance of missing the two crucial ones out of the many noncrucial ones.