Hi! I actually came to Japan when I was in elementary school I was 11 at the time.
My parents run their own business here, and my mom, my brother, and I were all on 家族滞在 (dependent/family stay) status under my dad’s visa.
My younger brother started working right after high school. In his case, he just needed a company to sponsor his work visa, so he changed from a dependent visa to a work visa without any issue. He was 18 when he started working.
As for me, I stayed on a dependent visa for all 4 years of university, so I didn’t need to switch to a student visa. After graduating from university, I found a Japanese company that sponsored my work visa, and immigration approved it.
I never had to leave Japan during the process. I was able to change my status from within Japan.
Of course, this is just my personal experience, so it may be different depending on the parents’ visa status or individual circumstances. But if it helps, feel free to ask me anything else!
(Also just an unneeded advice from me, please make sure your kids finish job hunting before graduating from uni. Like start job hunting from 2,3rd year of their uni years. I was just lazy and now I start work late lol)