I'm also a homeschooled student! I'm a senior and have gotten my appointment to West Point (still waiting to hear back from USAFA and USNA), so getting into a service academy as a homeschooler is definitely possible. Your stats look good as well, here are mine for comparison if it's at all helpful:
GPA: 3.97 (unweighted)
ACT: 33
SAT: 1420
Extracurriculars:
High School Girls Swim & Dive, Varsity, Team Captain (I'm a part of my district high school's swim team three months of the year)
Club Swim Team
Homeschool National Honor Society
President of Church Group
Co-founder: Creative Writing Club
Lifeguarding/Swim Coaching job
*not really an extracurricular but still good for my application: I've taken Dual Enrollment classes at a local university freshman through senior year, earning 31 credits*
I totally understand how finding extracurricular opportunities as a homeschooler is super hard, especially "school clubs" and such. The creative writing club I mentioned above I put as a "non-school club" in all of my academy/college applications, as it's something I started with a small group of friends to help each other become better writers. We had meetings in person or over zoom (COVID) and critiqued each other's stories and other writing pieces.
Tae Kwon Do is also a great sport for leadership (my sister did it for a while). Did you have to lead any volunteer opportunities as a part of it that you could speak to?
And as other people mentioned, maybe seeing if you could join a spring sport this year at your local high school would be a great opportunity. If you haven't already and are able to, also consider applying to boys/girls state (I attended last summer and that was great to put on applications), and are you going to apply for any of the summer programs at the service academies? Those were fantastic opportunities for me.
Finally, speaking to your current stats, your ACT score is pretty good, are you planning on taking it again? I took it four times and it was definitely worth it, especially regarding the math section. The first time I took it I got an 18 (!) but then studied and got a math tutor and got it up to a 28. I'm lucky to have had the resources for that but I think just taking it a few times is also good because you can build your confidence each time.
All in all, this was super long and I'm not sure if it's helpful. I understand how tricky it is to find extracurriculars and such as a homeschooler, but everything you have looks great so far and I'll be keeping my fingers crossed for you! Let me know if you have any questions about anything additional.
Edit: I forgot to mention two more things. Some super helpful advice I learned from admissions at West Point (Probably the same for the other service academies) is to get everything done in your portal as soon as possible!!! I can't stress that enough, especially DODMERB. I started the DODMERB process in the summer so thankfully was cleared by August, and that was a huge relief for me. Pushing deadlines is not fun!
Also, my congressional district selects a primary nominee and then sends an unranked list (as far as I know), to the service academies when it comes time for nominations. Acing the nomination interviews is huge! I was lucky enough to have a great interview with my congresswoman's board of interviewers and earned the primary nomination, which is definitely a large part of how I got my appointment to USMA. Even if you're in a district where your nomination source sends the service academies an unranked slate, practicing answers to interview questions and making sure you present yourself well is super important. Obviously that's a bit further down the road in the application process, but still very important I think.
