How do same sex couple get a divorce in the military if it's not recognized by the State?

While not exactly on point, due to gender, the folks at the National Center for Lesbian Rights, http://www.nclrights.org/ , work this issue all the time. What you will likely need to do is establish residency in a state that recognizes same-sex marriage. It is a case of the law not catching up with current events and people rushing to pass laws without thinking of second and third order effects.

If JAG is not a resource for whatever reason, look at these alternatives to try and find an answer. I would imagine, at minimum, you should be able to receive direction.

Many civil legal matters can be handled by a military legal assistance attorney at no cost to servicemembers. Civil matters include landlord-tenant disputes, wills and trust, family law (including divorce, separation, custody, and child support), naturalization, consumer issues (including debt collection, consumer scams, identity theft), powers of attorneys, advanced medical directives (living wills), and asserting your rights under the Servicemembers Civil Relief Act.

http://www.americanbar.org/portals/public_resources/aba_home_front.html

Legal Assistance Resource from the American Bar Association

http://www.americanbar.org/portals/public_resources/aba_home_front/information_center/working_with_lawyer/information_about_lawyers/military_legal_assistance/civil_matters.html

The ABA Military Pro Bono Project accepts case referrals from military attorneys on behalf of junior-enlisted, active-duty military personnel and their families with civil legal problems, and it places these cases with pro bono attorneys where the legal assistance is needed. The Project is also the platform for Operation Stand-By, through which military attorneys may seek attorney-to-attorney advice to further assist their servicemember clients.

http://www.militaryprobono.org/

JAG's will still provide guidance on what you can do. They will not represent you in court but they will help with any questions about the legalities and laws for your area. That being said....if the state you are currently domiciled in recognizes same-sex marriages....you can get divorced there. I hope this helps Shipmate. Let me know fi you have any questions or issues and I will be more than happy to help as best I can.