Petitions unlock partway into the second week by running into a new storyline NPC, The Activist, outside Blodgett Gym (contributed by noonycurt)
Current petitions include: make the gym lockers unisex, allow swimwear to be considered proper dress for classes etc., allow everybody to be topless wherever men are currently allowed to be topless, disallow anybody from going topless (contributed by fhish and noonycurt)
You can't start right from the topless petition.