I think there are at least three: hemisphere, section of an obloid, cone.
Since you did not say face, you appreciate difference between definitions in platonic solids as defined by Euclid and Euler, and the curved solids. A hemisphere has one flat surface and no vertices, but so does a cone. A vertex is defined as the meeting of edges, which are defined as straight in euclidean geometry. Since there are no straight edges coming to a point in a cone (unless you want to talk about infinite edges emanating from the flat surface), there are no vertices on a cone.