Are there systems or processes that gain an advantage by operating in the vicinity of absolute zero? Any related to Charles' Law?
There are quite a few applications of reducing temperatures to the vicinity of absolute zero. Though I do not think there would be any application of doing that in the study of Charles' Law. The relevance of absolute zero with Charles' Law is the fact that it defines a temperature below which temperatures cannot fall as at any temperature below absolute zero the volume of a gas would have to be negative which is not possible.
Low temperatures, neighboring absolute zero have helped in the study of superconductivity- a state where the resistance of metals drops to zero. At this state, a magnetic field can be maintained forever with very little power usage. This can be used in making magnetically levitated trains and the alike.
Another phenomenon that is exhibited at extremely low temperatures is superfluidity.