Is it possible to do independent research and publish a paper?
And how would one publish said paper?
There are programs that allow high school students to do independent research that might lead to publication. You can look for them on the homepage of mysummeradvisor.com. There is an article. Many of these give a student a faculty mentor and use of lab facilities.
Publication is a high bar! You should look up the Journal of Young Investigators (I think that’s the name). They focus on publishing work by young (typically undergraduate level) students.