I heard that the car battery cannot supply current at a fast enough rate
(because of chemical reaction neccesary)

You heard wrong. Current due to electron flow moves at the speed of light.

for subwoofers and that a cap is
needed.

At what power ratings should one start to worry about this?

You should start to address whether or not a capacitor will help you if your
headlights are dimming such that it's become an annoyance to you. When that
happens, consider installing capacitors as close to the headlights as
possible. Oftentimes it won't be much more money to buy a high output
alternator or have yours modified by a shop. This is a much more effective
tool to reduce dimming.