What is the term for a person who conducts illegal business? Surely it is not illegal businessman, but maybe that could work.
It will be in any case a criminal, maybe a career criminal. When it goes to drug business, you call it a druglord, which in himself is a kind of kingpin. You can choose your poison.
But in the end, without contest, the word which would best fit your definition is a racketeer, although career criminal or illegal businessman are not bad themselves.
is someone who commits crimes involving business activities, mostly desk-work crimes like accounting or insurance fraud.