Running an integrated speaker amplifier board off DC?
Since you already have a port marked 12 V DC ...you have your answer, Yes you can use a DC battery supply.
Simply use the wires to that port, or connect to the connector location on the PCB to use the amplifier battery powered.

Since the transformer output is 12 V AC, the amplifier is likely able to work up to 15 V DC or more, but this is where you need to be just a little careful. A fully charged auto battery is about 13.8 V, and a 3S LiIon will be  about 13 V ...both well within range. Just be aware that some batteries if on a charger the terminal voltage may rise above what you expect.