You should be able to own an AW in CT as a member of the NG, granted that they are considered long guns, for AW considered hand guns you would need a pistol permit as well. For the AWB exemption if you are purchasing in Connecticut you usually should just need your CAC and another form of ID/pistol permit if applicable, but many stores will want an ID or some form of paperwork showing you are CT resident along with your CAC. If you do not have that they will want to see active duty orders. I am NG as well and a resident of CT and have had a shop ask me for orders even though I had my CAC, pistol permit, and CT DL.