How can I get an internship or research opportunity at a hospital/lab if I'm not 16?

I would say to not go the traditional route and apply for internships. Directly contact the head lab or the person that manages it. If you have any events at your school which include introducing researchers, make sure you get their contact so you can ask personally for an opportunity. I currently work a Chemical Engineering lab, and I'm 16 currently. I work with mice brains and its awesome. Last year, I worked at a genome lab but worked more on multimedia. These internships are from searching the contacts of researchers from a university website, asking to shadow them or directly for an internship. Its really putting the first step into the door. Hospitals are very difficult as you have to be 18. Hope you get an awesome internship in the future.

at this point, I would probably look for opportunities to shadow doctors or other medical professionals in local private practices; that might be more accessible to you until you're 16, and the closest thing to hospital/lab internships if you're interested in medicine. there really isn't some kind of workaround for this—the reason hospitals and labs will require you to be 16 is basically labor laws that restrict the number of hours people under 16/18 can work. so you are going to have to wait a bit until you can apply for those really intense internships.

(i'd also say that it's really rare to find people doing prestigious internships before 11th grade, outside of a couple of really specific super super competitive magnet high schools. compared to the vast majority of students, that wouldn't put you behind at all)