Looking up the relay model number: "SRD-12VDC-SL-C" it turns out the the 'L' in 'SL' means the coil draws 0.36 W. If that 'L' were a 'D' it would mean 0.45 W.
0.36 W at 12 V is 0.030 A (30 mA). 30 * 16 = 480 mA so your 12 V supply should provide at least 500 mA just for the relays.
The LEDs are powered by a 5V regulator running off the 12V supply so you probably need to add at least another 160 mA for those.
12 V 750 mA would probably be sufficient but 12 V 1 A would have more margin.