While it would be great if you could get a chance to work with a professor, understand your chances are slim. Professors have their own grad students and undergrads to supervise, as well as teaching classes and doing their own research (and applying for grants and writing papers and serving on committees...). It's often hard for a undergrad to get a chance to work with a professor - they don't have as much time or background in the field as a grad student, so they take up a lot more of the professors time for less result. A high school student would take up even more time, in theory, and require a lot more training. Many undergrads will start out working in a lab on someone else's research project - defining your own project you can do in a semester or two with original results may be unfeasible.
You might have a bit more luck contacting grad students than professors - some of them are capable of advising you in a research project and might be able to 'outsource' some of their research to you. But this is just speculation - as a grad student, I'm helping my adviser oversee a few undergrad research projects. And yes, it's taking up a lot of my time that would be better spent on my dissertation - but it will look good on my resume when applying for faculty jobs. Good luck.