The World Zionist Congress, founded by Theodor Herzl ym’s, has no influence on normal Yidden.
Mishpacha is on the left-wing of the chareidi world. Forget the position of Rabbonim or Roshei Yeshivos as to voting in WZC elections, none of the gedolim do so, but rather ask the more pertinenet and unanswerable question as to which Rabbonim and Roshei Yeshivos support Mishpacha.