Is there a way to review weapon system qualification scores aside from the Enlisted Record Brief?

Your training office in both the Company and Battalion should have all training records available. In the Battalion that with writhin the S3. Thank you for your service.

Your unit and or higher echelon's training NCO should have those scores in DTMS

DTMS!!! Used to be called "My Training" tab in the old AKO, now it's some sort of card? Either way, training tab in ako and you should see the dtms button.
Then there should also be a score in your Army Career Tracker at actnow.army.mil.
Your leadership should also have a hard copy of your last weapon qual.