Having worked in a number of research labs, I would add to this that you should also try contacting professors at the nearest college. Very rarely is there something structured put in place for high school students to work at universities, but that said, many professors will work around/through/over the bureaucracy so that you can work with them. Often times all that is needed is a bit of initiative and some personal contact.
In other words, if there's one thing I've learned so far in life, it's not to underestimate the power of simple human contact. Go knock on a random professor's door, and I think you'd be surprised how many times the door opens!
Very rarely is there something structured put in place for high school students to work at universities...
Really? I went to three different summer programs at Ohio State University when I was in high school. Great fun.
My last summer program ran nine or ten weeks and had me playing RA in a biochemistry lab. One of the most generous masters' students in the whole world (I think the poor guy was in his 3rd or 4th year... he was kind of a long-term masters student ;) spent most of the summer teaching me stuff -- growing bacteria, lysing them, running chromatography columns, gels, enzyme activity assays, the works.
He was crazy, actually, to spend that much time teaching a high school junior who would never come back to the lab, but I hope he's having a happy career somewhere because he sure was generous.
Large universities (and often small private universities with an interest in being linked with the surrounding community) are more likely to have structured programs available. Smaller state schools less so. Even universities which do have such programs will often not spread them evenly to all disciplines. I'm also coming from a biochemistry background, and because biochemistry tends to be labor intensive I know programs are much more likely to exist in this area than, say, computer science or math.
Obviously, if an organized program exists, take it! My point was that the absence of such a program does not preclude you from taking an individual initiative to do something creative.
The Intel Science Talent Search is pretty much exactly like this. You work on graduate-level research with a university professor as your mentor.
Even if you don't care about the competition, it's an easy pretense for getting in touch and working with a professor. Plus, you might be able to convince your school to give you time during the day to work on your research (when I was in HS, we got a period every other day).