At 12, I started working on an open-source Ruby gem in a not-so-niche market (would rather not say, might give it away). I (American) had an offer from overseas (Britain) to work on a very small project in Ruby using the gem I created. I got paid about $200 I think, over PayPal. That made me want to start working.
Someone reached out to me right after my 13th birthday, saying they wanted to hire me to work on a Rails application, and their idea seemed good. I got nervous when the guy (whose LinkedIn seemed legit) sent over an NDA, but I figured, how much trouble can a 13-year-old really get in? I didn't even tell my parents. I was basically getting paid to work on my gem, which I would have been doing already - no big deal. After about 2 months, I think the guy caught on to my age or lack of experience, because he told me I had finished what needed to be finished. All in all, I made $3500 for about 60 hours of work.
I continued working on my gem for free after that, and other companies reached out for internships, etc. I turned all but one down, because school had started in the fall, and that turned out to be only two months or so of an unpaid internship. I dabbled in other side projects after that - I was bored of working on the same project for what was now a little over a year.
Imagine my surprise when I come back just 6 months after that, and there are two forks of my gem on GitHub and it's been downloaded 30,000+ times (not even counting the other two forks). And yet, the work offers had dried up. I didn't have a website, no recruiters had my contact information, and I was a little lost on what to do. I volunteered that summer, telling myself that more opportunities would arise.
I haven't worked on that project in months. I'm not sure if I'm burnt out from programming (haven't done much of that recently either), but I'm definitely burnt out from that project. I logged onto RubyGems recently and between the original gem and all the forks of it, it's been downloaded 100k+ times. If I created a blog, and started updating the original gem again, I'm sure I could find some work. But I'm not interested in doing that, and I'm not sure why.
I'm headed off to college in the fall internationally, with less work experience than I would've liked and less money in my pocket, too. I'm not sure what the lesson from my story is, but hopefully it will make you take the "too fast, too soon" ideas to heart. Don't stop coding - if it's what you love. But don't let high school pass you by either (just read that you're actually in middle school). My best memories aren't receiving my checks in the mail or making a résumé or even coding, but the time I spent with my friends.
To answer your question directly, no - I was never successful in finding advertised internships for young people. It was discouraging. I'm probably lucky that guy who paid me for months didn't confront me about my age - I'm not sure what would've happened.