Don't give up on your dream
Don't give up on your dream. You absolutely have the ability to gain acceptance to West Point. I'm a senior this year and was homeschooled most of high school and was accepted to USNA and am on the NWL for WP because I didn't win the vacancy for my Senatorial nomination. I also have three friends who were homeschooled who got into West Point this admission cycle.
When you are homeschooled you have a lot more opportunities for extra curricular activities. I ran cross country with a homeschooled team. I did five years of debate (there are homeschooled leagues - NCFCA, Stoa - as well as the public league - NFL) and then went to state in mock trial and quarterfinals at the national moot court championship. I did a ton of forensics related stuff and the list goes on, but it was an area where I could demonstrate leadership in place of things like getting student body president.
If you still have time, try to become an eagle scout. I didn't, but I feel it definitely helps.
As a homeschooler I was able to take dual credit classes at my local community college for free. You also CAN take APs as a homeschooler. You need to call around to local schools and they will allow you to take APs with their students. I'm graduating with 7 AP's and 26 hours of college credits.
I also decided I wanted to attend a service academy during my sophmore year and I looked for ways where I could differentiate myself and being a recruited athlete wasn't going to be one of those. I'm in good shape and am a runner but I'm nowhere close to being competitive at the collegiate level. So I started taking Arabic my junior year in the evenings at college and for my senior year, got a scholarship to attend a top boarding school in Jordan for Arabic immersion (which is why I've been homeschooled throughout most of high school and not all of it). I think my Arabic proficiency definitely helped me in the admissions process as I can get by pretty fluently.
Don't let being homeschooled dampen your dream. You're in a different situation but that's not a bad thing. If you make use of every opportunity available to you and work to create new ones, your dream is a very realistic goal. If you want any information on things that I did or want me to put you in contact with some of my other friends who got into West Point this year I'd be glad to.