As some comments suggests, it's not clear what you want from the professor. If there's no clarity from your end, it's very unlikely the professor can help you (even if you get a chance to meet them). I recall my thesis adviser saying something like, "I receive >100 emails/day from various students requesting to work with me". This was in 2001-2002.

I "hustled" few professors and got some face time around 2001 time frame. But I was out of undergrad (not high school as you requested). These are the things I recommend.

0) Have clear goals on what you want from the professor (and what you can offer too). I showed interest and competence in doing research and I wanted funding from them.

1) I worked extensively in couple of areas (Genetic Algorithms, Simulated Annealing, Finite Element Methods, Optimization of Manufacturing Processes) and published conference and journal papers before requesting a meeting. I am not suggesting you do this. But you need to show you have done some work to warrant face time with professors.

2) After you have done 0 and 1, email them, call them and somehow "stalk" them "respectfully". To meet the professor I did Master's thesis, I wrote to him few times and called him. But finally, I waited in front of his office few hours every day and finally got a chance to meet him. I eventually got funding and completed my degree with him.

3) Alternately, if you can attend some conference (figure out a way not to pay but attend the conference) and meet them there. That's a big plus, IMO.

4) Write to the professor's PhD students and start communicating with them. Learn from them and do your research and build some projects/ideas and use it to meet the professor. This is a "back door" entry in to professor's lab. But PhD students are quite helpful and can guide you in to building a "portfolio" of projects.

HTH.