Where can I inquire about storage of personal weapons at Fort Jackson?

Contact Base PMO for Information...

1st: As you are a not assigned as permanent party, you are a transient SM at Jackson...I think you will find policy on weapons in the school house polices..or at least there used to be..

As a student and not assigned as permanent party, Id bet you will not be allowed to store them in the unit arms room.

As you do not reside in family housing, and post policy #25 states they Must be stored in the "appropriate location, contact your unit 1SG"
http://usasd.armylive.dodlive.mil/files/2016/12/USASD-Policy-Memorandum-21-26.pdf

I think you will find no on post solution. You may find policy that forbids a SM in student status from having one at all....
If no such policy, then consider storing off post. Perhaps in a fellow SM's privet residence, as suggested..get the smallest storage locker you can at a privet, reasonable safe facility. Get your self some "Damp Rid" store the weapon in a small unmarked, container, locked, with damp rid . Do not advertise the weapon.... do not let others see you go to storage locker with boxes of ammo, or a Cabalas bag, Academy bag, or the like.
Get your self a "bag" indicating something of little value .. Food store reusable bag, sears bag, something no one would look twice at, and carry weapons and weapons items to and from storage location in that.

So unless you want to be told NO, as that is the easy answer most will give, vice the harder ...well this is how it COULD be done..
Call the school house..and ask for a copy of all policies and guidelines DO NOT ASK can I bring a weapon.

Call with PMO and ask for current POW polices and guidelines for both those assigned, and those in student status.
Read everything TWICE and consider if there is policy that directly forbids your possession of a POW whiles assigned to school. If not, seek a off post, out of sight , out of mind solution.

Not sure what is at or around Jackson anymore, as I have not been there since 2004, but I know that other installations have a Rod and Gun Club that offers storage, but the weapon needs to be registered on post and you are at the mercy of their operating hours. Also, some gun shops/shooting ranges offer storage options without on post registration, but once again, you are at the mercy of their operating hours. If you are authorized to live off post then maintain the weapon in your residence, and if living on post the unit arms room would be the answer, but as SFC Smith said, check with the PMO, as well as the schoolhouse for their policy.