The question is rather vague. On a two layer board the use of planes for Vcc/-Vcc may be overkill and may add unneeded complexity to the design. A simple 12 mil trace on 1 oz copper will carry an amp according to the calculator I just popped it into.
Since I am starting with KiCAD, I thought that it was required to have filled zones in the PCB layout.
If I understood correctly, I can design a 12 mil trace (or another sufficiently high value) for +Vcc and -Vcc. Thus, it would be necessary to have only the GND plane in both layers?
There is no requirement to have filled zones unless your design requires them. That goes for ground planes also. Generally they don’t hurt and may even be helpful in simple designs and people recommend you generally use them.