In the State of Washington, I don't think that there is any problem with kids taking the ASVAB.
Both of my sons were home-schooled, and both went to the local High School and took the ASVAB when in their 10th grade year. (The older boy got super high scores, and the recruiters were sorely disappointed to discover he was only 14.)
But as remarked elsewhere, a semester of college courses can alleviate any concerns about the academic record of a home-schooler. Another family I know had 2 children join the Marines, and both are doing fine. Both were home-schooled through 12th grade, and both did about 1 year of college courses at the local community college.
Sometimes, it might be necessary to get the GED. My daughter was offered a 4 year ROTC Scholarship (Navy, nursing program), but the 4 year college was willing to accept her if she did the GED (a minor condition, since she had 2 quarters of college courses.)
Parents can prepare a "high school transcript" ... and we had no problems with the transcript submitted for our older boy, when he started at the Naval Academy. (Of course, since he had 2 years of college courses via Running Start for his 11th and 12th grade year, and had a 3.93 GPA - with heavy math/physics and chem. courses ... no one could accuse the parents of hyping his high school grades when his college grades looked pretty good!)
Bottom line ... there are ways to work the system!! The biggest thing to remember is for the prospective recruit to get as much information to ensure he/she finds the best fit/program to match interests/desires and skills. Good SAT scores, good ASVAB scores, and perhaps some college courses will all help make an attractive package.
R/ Mike
Excellent post. You must be *very* proud. :o)