If your system will tolerate a range of voltages rather than exactly 12v, then the answer would be yes.
You would need two batteries, one with it's negative terminal connected to ground to provide the "+12v" supply and the other with it's positive terminal connected to ground to provide the "-12v" supply.
It would be a good idea to place fuses right at the battery terminals to protect against accidental short circuits and other faults.
Also, liquid-electrolyte lead-acid batteries require a degree of care in application and may not be the best choice outside of vehicle/marine applications. NiMH cells or sealed gel electrolyte batteries are generally preferable for smaller systems, especially if used indoors.
Finally note that this is only straightforward if your amplifier is only connected to the batteries when they are not being charged. If you remain connected while they are charging (particularly off an engine) additional protection circuitry will be required.
An alternative course of action would be to use one battery and a dual output DC/DC converter. This would also get you a more consistent output voltage. Or you could re-design the circuit to operate from a single supply, using for example a car-audio audio power amp chip.