Yes a writ petition can be filled by any individual or by an organisation or by a group of people in the Supreme Court under Article 32 of the Indian Constitution or in the High Court under Article 226 of the Indian Constitution if the fundamental rights conferred to the citizens by the Constitution has been violated. A writ petition can be filled by the victim himself or any person who is related to the victim or any social organization or a social activist.
A lawyer or any agent of law is not necessary for filling writ petition. It is filed if any of the fundamental rights as guaranteed by the Indian Constitution to the citizen is violated. Majority of writ petitions are filed against death sentences that are given by the lower courts to a convict as it violates the fundamental right of Right to life as per article 21 of the Constitution. It can only be filled in the High Court and the Supreme Court and not in the lower courts.