I'm not totally sure what you mean by a flat Earth map. The UN flag contains the UN emblem, which is a polar azimuthal projection of the globe extending to about 60 degrees South latitude, which shows all the landmasses on Earth except for Antarctica (which has a notable lack of permanent residents and thus representation in the UN), surrounded by olive branches.

The original logo, designed in some haste before the first UN conference in 1945 in San Francisco, also had a polar projection with the US in the center, but cut off parts of South America. The current design moves the center of the map to the North Pole, and rotates the map such that the prime meridian and international date line run vertically on the map (which also moves North America out of the center).

There are various ways of projecting the earth, which is basically spherical, onto a flat map -- all of which skew the sizes of different landmasses. The difficulty of picking a map that would represent an international organization is that the "traditional" Earth maps that have the Americas on the left, and Africa/Eurasia/Australia on the right, is that they tend to center the world on Europe or at least the Atlantic, and understate the size of the Pacific because it's cut off in the middle. The polar projection is a neat workaround for that.