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.