I got a number of internships in high school by just cold emailing a TON of startups, and making it very clear that even though I didn't have a ton of experience, I loved programming and learned quickly. I had a low success rate -- but I really only needed one person to say yes!

I recommend emailing the highest-up person in each company whose email address you can find (or guess and check, via a service like https://email-checker.net/). Also, don't constrain yourself to advertised positions -- I never applied for an internship for which there was a listing.