Hi! The most important thing is that it is obvious that you care for and want the best for your own children or you wouldn't worry about it like you are. That's step 1 After that I would say to try for a little consistency, going in and out can be hard on your kids and on you as well. Moving a lot as a military family can also be difficult because your kids are changing schools so frequently. That in itself can be hard on them. I have several military families who have turned to homeschooling just to provide an environment of comfort and security for their kiddos.
It sounds like you and your spouse need to take some time to pray over the decision, and then decide. I wouldn't base my decision on if whether your family would support you at this point. Chances are they aren't close to you geographically if you move a lot anyway. But there will probably be other families where you go that are in your same situation and you can find support there. Do what is right for your family regardless of what others think or say.
All that to say that if you feel you're being called to homeschool, then I would go for it. If you are committed to it, then you can do it! And I do think it would provide some consistency for your kids as well. I'm not sure why you pulled them our, or put them in public school to begin with, but from just what I've read from your comments, i don't see why you wouldn't be able to start homeschooling when you get to your new state and make a clean start there.
I do have a post on transitioning from public school to homeschool here if it helps any:
Hope that helps Erica