Why is the Vatican a place of pilgrimage?
While religious sites are assumed to be the objective of a pilgrimage it can also be used in other conotations. A pilgrimage assumes a great moral search, or at least the pilgrim is taking the journey to enlighten themselves or out of a sense of reverence. You can make a pilgrimage to "Strawberry Fields", "Graceland", "DeLay Plaza" or the "Vietnam Memorial Wall". It doesn't have to be to Jerusalem or Mecca or the Vatican.
Lourdes is a pilgrimage site for our Lady in the south of France. People fast and go without sleep while on pilgrimage in Lough Derg in Ireland. Fatima is a pilgrimage site for our Lady in Portugal. Many people go in the walk for the pilgrimage to Santiago de Compostela in Spain. Vatican City is a place of pilgrimage as it is where the head of the Catholic Church lives.