Where would I even begin if I were to apply to a university during, day, my sophomore year of HS? I’ve heard that this has been done before, but how?
Bard collège in NY has early college program.
Also you can do dual enrollment at your local CC through your HS (for free!)
Here’s an idea: look into attending a United World College (UWC). These are two-year IB (high school) programs that you could attend your junior and senior years. They are FREE for US students, including tuition, books, room, and board. There are about a dozen and a half campuses around the world, including one in a grand old railroad hotel in Montezuma, NM. You attend with students from all over the world. At Montezuma, 1/4 of the students are US students. At the other schools (where about 1/4, usually, are also from the host country), US students usually number about 4-8 in student populations of about 200. Students are often top students from their countries.
These are top international high schools where students often go off to top universities, in the US and elsewhere.
As a US student, you actually apply for the full scholarship (Davis Scholarship). If you receive that, you pref schools and are slotted to one of them. Close to half of US students go to New Mexico, the rest overseas.
This might be a great way to have a bold, transformative learning experience without going off to college first. I do want to emphasize that these schools really adhere to their mission statement. Read the mission statement carefully and see if that resonates with you. If it doesn’t, that’s okay. But admission will be very difficult. Academics are important to admission, but being in sync with the mission and vibe of the schools and their purpose is much more important. Good luck!
UWCs are an amazing option - however I wanted to clarify several points that you brought up. I’ll be attending UWC Swaziland in January. The scholarships offered by UWC are need based - depending on your parents’ income bracket, you could end up paying anything from the full tuition which can reach $40,000 or receiving full scholarship. Applicants are also ranked based on the interview process, and their scholarship fund occasionally runs out of money if there’s an applicant that they would rather put the money towards. The UWC experience is an amazing movement, however competition is extremely intense. That being said, they are a great choice if you’re looking for an alternative education! 
Clarkson U has an early college program as well.
@CharlieCat, Congrats! Have a great time at UWC Swaziland. Are you attending through the US committee? When the student I know attended about a half dozen years ago, all US students attended without cost, regardless of income. I know this to be absolutely true. Do you know if that’s changed? Thank you.
Thank you! I’m actually attending through the Canadian committee - and my apologies, I double checked and the US scholars receive Davis scholarships. In most countries the scholarships are need based and I believed the US committee operated similarly. I think in other countries there’s more spots available because some students have the option of paying full tuition, however in the US only fifty are selected to attend.