High school summer research in fields other than biology/medicine
This is not really a proper answer, but rather more like a stretched out comment. In the field of biology, lab hands can be relatively quickly trained. Now, they probably won't understand much of the science that is going on behind the scenes, but doing some of the routine tasks of "pressing buttons", they can free the more senior researchers to do the more challenging parts of the research. 

Mathematics, on the other hand, often requires several years of training before you can start contributing in any way and there is no good way to avoid this. A beginning university student might have heard of the concepts that are being researched in some lab, but it is unlikely that a typical high school student can do much. This is all something that you already mentioned in your question. 

Having said this, many things in the field of computational science (be it physics, mathematics, astronomy, biology, sociology, or whatever) are rather mundane and could easily be done by someone with a limited knowledge of the field (and I remember there being a recent example in astronomy where a high school student coded up an algorithm to find some shapes of galaxies/starts/something, and it worked so remarkably well that it ended up being published, but I could not find the reference right now). If you truly are motivated enough to contact and seek advisers in your area yourself, I am sure that many would be happy to show you the ropes. Don't be too discouraged, though, if some refuse (or do not even respond), as they might be too busy doing their own research to train new people who will shortly be leaving the lab, anyway.

EDIT to reflect the revised question: I want to point out that earlier I suggested computational science, but there are also experimental physics (for example) labs, which might also have something to do for an interested high school student. 

There is no separate department for computational science, because it is not really a study of its own; rather you can use computers in any field (for the sake of completeness, there sometimes might be a department called computational science, but its scope is usually more limited than my definition here). Suppose you were interested in physics at institute X. Go to the website of the department of physics of X and see what the groups are working on. Find the homepages of the individual groups (professors) and see if they seem to be using computers (it will usually be more or less obvious from the types of figures on the website, or they might just explicitly mention it) and if their research appears to be of interest to you. See their publications, and while you some of the papers might be behind a paywall, try Googling the titles and you might find them on arXiv.org or similar free to access preprint servers. 

Now the publications themselves are often quite technical, don't get disappointed if you don't understand much, but the introduction section should often be more or less understandable. One of the things you might want to look for is to see whether the group is using some software package (they should cite it if they do), or if everything seems custom made. In the former case, a considerable amount of the work might be in preparing files for the software and with some training could easily be done by a high school student. 

As for approaching a professor, and I am hoping that other users on SE might comment more on this topic, the best way, I think, is to have read a paper or two by the group and to point out in your email that you have a rudimentary understanding of the science and the scope of the work. I, and I am not even the group leader, get emails from students (often from other countries) looking for a position (bachelor's thesis or something similar) at our lab because they are 'engineers'. Being an 'engineer' hardly qualifies them for the rather specific work we do and shows that they have not done any research as to who to send the email to, nor to what the group is working on (as they never, ever, specify what they want to do or why they want to join our group, in particular). These types of emails probably never get a reply from anyone (although a high school student just might, for their lack of knowledge as to the specifics would be more understandable). Essentially the more specific you can get in showing your knowledge and asking some questions, the better. 

Much of what I write about is a lot to ask from a high school student, and might not even fully pay off in the end (i.e. you might get rejected). I hope that in any case this gets you interested in reading academic papers and gives you a better understanding of what it is that scientists do.