It is definitely possible, but it depends on your learning style. For a new course you would have to educate yourself through textbooks, online resources, videos etc and it would depend if you find it easy to learn this way, also without any guidance from a teacher. You would have to test yourself with past papers to track your progress and also to familiarise yourself with the exams. Looking at mark schemes and exemplar responses can help you learn exam technique.
It can also be a bit pricey as you have to pay for textbooks and the exam fees (per exam). It would be helpful to contact an exam centre in advance to see if they are willing to accept you and if they can take your chosen subjects and exam boards. Some exam centres listed on exam boards websites sometimes reject private candidates even though they're listed as a private exam centre. It also depends if your subject has coursework, as usually this is marked by a teacher. Some subjects have an alternative to coursework paper.