I don't believe that the government or a school can force any student to get vaccinated, nor do I think that they will try.
However, students who choose not to get vaccinated may still have consequences to accept. For example, when the HSC exams are held, non-vaccinated students may need to be kept separated from the rest in doing the exams (perhaps in a different location or further distanced). Perhaps the rulings on masks will differ.
No student will be prevented from taking HSC exams irregardless of vaccination status unless something really dramatic happens. However, any student who contracts COVID-19 risks being physically unable to take exams and potentially a long-term health problem. Vaccinations have very substantial protective effect against serious illness and hospitalisation, etc.
I strongly encourage everyone to take up the opportunity to be vaccinated and for those considering declining, to ask yourself whether your reasons and sound and wise. For example, I have a serious allergy to certain preservatives and I would not take a vaccination that contained one of them - it could be fatal for me. FYI, I have checked and neither of the present vaccinations are problematic in this regard for me. My point is that there are potentially solid reasons for not taking a vaccination, and that anyone considering not being vaccinated should reflect on their reasons.