I have tried cold-emailing various large and small companies asking for an unpaid internship placement with my résumé attached.
For a high school student with no university experience, I just don't think this approach is going to work well for you. As someone young and inexperienced, companies would be taking a huge risk hiring you. Even university interns don't contribute much to the company, but the benefit is that they are very likely to come back to work for the company in the future and will already be trained. With a high school student, you will take a lot of training and mentoring, and there is much less likelihood you will want to work for them in 6-8 years, if you're even in the same field. You are an unknown entity with little potential benefit for the company, so cold-calling is probably not going to be successful.
Another reason you probably haven't heard back is that you are already partway into your summer break. You're coming to this late in the game, and companies aren't hiring interns anymore. If you are already looking for next summer, you're actually a little early and should probably wait until the fall.
However, all of this isn't to say that you shouldn't be still be looking for an internship! There are just better approaches you should take. You either need to go someplace that already has a reason to trust you despite your age, or go someplace that you know is specifically looking for people of your age and experience.
You are most likely to have luck (and maybe even for this summer) going through family and friends who know you. It can be a computer store you visit frequently, a friend's mom, your neighbor, or your physics teacher's college buddy. You need someone who knows you and your work ethic and can vouch for you. If you do manage to find someone who's willing to let you come and apprentice, don't expect to get paid. Like I said before - you're going to be doing much more learning than actually contributing.
Another option, and in my opinion the right way to do this, is to seek out internships that are specifically intended for high-schoolers. Some companies, government labs, and in particular universities have programs focused on teaching and educating students. Start searching for opportunities in your area specifically for your age group, and talk to your teachers and guidance counselors to see if they know of any programs. There probably aren't a lot of internships like this, but you probably also won't have as much competition to deal with when applying either.
