Who has written the law of karma?

That’s like asking - who has written the law of gravity?

Karma is a natural law of the universe, and just like the universe is created by God, the laws of the universe, including gravity and karma, are also created by Him.

As far as writing goes, though, Chitragupta is the scribe of the king of death, Yamaraja, and he notes down all the sinful activities of every single human in the world, by which a person will be judged at the moment of death.

In his famous work, Sri Padyavali, Srila Rupa Goswami quotes:

O Supreme Personality of Godhead, when someone desires to chant Your holy name, sins tremble in fear, the glory of material illusion faints unconscious, Yamaraja's scribe Chitragupta becomes happy and gazes at the chanter's toenails with awe and reverence, and Lord Brahma prepares madhu-parka to worship him. O Lord, what more can we say than this? -- author unknown

—Sri Padyavali 20