It's kind of subject to the local garrison commander's preferences, but a lot of bases do let permanent party in family housing keep weapons in their homes, however there is strict registration and you may be ordered to intern them at the arms room at any time at what amounts to the whim of your unit commander. You also can't take them out of the quarters except to go directly to or from an approved use like hunting, recreational range shooting if there is a range handy, repairs, or sale/purchase.
Troops going through deployment processing may or may not have their assigned personal weapon with them, depending on the mob station and the means of transportation that will move them on from there, but they absolutely won't have any ammo for those weapons on them during the processing.