There is an "extension" to .obj where vertex colors are added to the v line, like

  v x y z r g b

But this stores a color per-vertex. Vertex colors in Blender, like UVs, are per-corner-of-poly. The ideal way to store vertex color information in .obj would really be

  vc r g b
  p v1/vt1/vn1/vc1 ...

As it is, exporting per-vertex colors would require splitting a vertex into multiple vertices, one per distinct vertex color that it uses.