Who do I need to talk to about getting reimbursement for my damaged items after my room flooded?

That's why you're supposed to have renters insurance.

So, yes, renter's insurance is the best answer. In any of the below scenarios, renter's insurance SHOULD cover you, and then THEY can file a claim against a responsible party (if any).

However....

If the room was flooded due to a fault in the barracks (for instance a pipe burst) then the claims office (same office you go to when mover's break your stuff during a household goods shipment) SHOULD be able to provide some relief. You would, at that point, essentially be filing a claim against the government for the damage caused by their faulty structure / faulty maintenance.

If, however, the flood was due to an idiot roommate or neighbor, your claim would be against them, and you would be asking them to make you whole, either person to person or through small claims court

If the flooding was cause by an act of God (i.e. actual flooding due to heavy rains) you have only renter's insurance to fall back on. And that may or may not be helpful, depending on the terms of the policy.

Renters insurance. The Army isn't your insurance agency and isn't going to reimburse you