Looks like you have a maximum current consumption of 10 x 20mA per colour per board, total 200mA (which is just fine for a BC337 to switch). Times 3 for the colours and you have 600mA total per board. Multiply that by 14 and you get 8.4A. So yes, you will need a high current power supply. One possibility is a standard ATX computer supply.
It would be easier to use higher voltage and lower current. If the LEDs drop 3.2v each, then instead of 10 led/resistor combinations in parallel, you could have 2 sets of (5 leds in series + one resistor). This would drop around 16v, so you would need (say) a 20v supply to drive it, and drop 4v in the series resistor. But it would consume 40mA per colour per board instead of 200mA. That brings the total power requirement to 1.68A at 20v, which is more manageable than 5v 8.4A.
Another option is 5 parallel groups of 2 leds in series, which would need about 9v @ 4.2A. If your boards had 12 leds instead of 10, you would have even more options, i.e. 12x1, 6x2, 3x4, 4x3.