This shouldn't be impossible, it will just require some initiative and hard work - which is exactly what I would think the service academies will be looking for.
It will be hard work either way - but it is encouraging that they have an entire Web page devoted to homeschool applicants now.
We did a tour of Virginia's Senate and got to chat with the local reps. Running here and there throughout the sacred halls were the cutest little high school students in blue blazers. Pages, all of them volunteer, many of them homeschoolers. Asking your local representative if your child can work in his or her office is a GREAT idea to gain experience, an excellent activity to list on an application, and a source for impressive recommendations.