Can you own a stab proof vest in Australia?

In NSW, yes. The Prohibited Weapons Act in this state prohibits the ownership and use of a ballistic (bullet) proof vests without a permit; however, stab vests do not come under the definition as described in that Act. Be mindfull that customs have ammended their import regulations to include vests that are designed to protect against a weapon, including a knife, so you would not be able to import a vest into Australia without the proper permits.