What types of volunteering can I do to show my interest in engineering as someone who has just finished year 12 but unfortunately has really no work experience at all except in year 10 at an aged care centre?
This one is quite simple. Because the answer is
any form of volunteering will look good

Restricting yourself to specifics doesn't always do you a favour. Volunteering in any way is nice because from different volunteering opportunity you gather different skills and traits (that of which, others might not have).
I originally wanted to do a major in software engineering but then I'm starting to think maybe telecomm or biomedical instead.
For something such as biomedical you won't need too much on top of what's done in 2U. To make your life easier you might like to consider certain parts of 3U (e.g. harder growth and decay) for extra revision. The thing about most engineering disciplines is that they make you do math in first year, which can be effort, but then you don't really need it later on.
(A bit of foundational statistics might be helpful, but keep in mind this is not taught in HSC 3U/4U.)
I'd imagine telecommunications wouldn't be too far away from what Jamon's doing though, i.e. electrical. For that one, whilst the actual concepts in HSC 3U math are quite useless, you need to be strong with mathematical ability as a whole. 3U+4U integration would be a nice boost. (I'm not sure if 4U complex numbers is really beneficial to telecommunications though.)
Those are just some ideas. But like, you shouldn't really be spending too much of the holidays worrying about that either. So if you're motivated to do some self-learning, do it, but don't pressure yourself into it!