This is essentially a (cw) reference request, because it seems like the sort of thing that should have been looked at already.
Setting aside, for now, how to think what the localization of a general space is really; what's the right way to think about localization of $SO(n)$ --- or, if that doesn't make sense, of $Spin(n)$ --- as a space? (whether Sulivan's construction or Bousfield-Kan or ...)
some points of curiosity: Is it still a homotopy group? Is it better known as something else? Is there any good interaction between the algebraic side of localization and the group structure?