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.
crook suggests the idea of illegal business better than
criminal; a criminal is anyone who commits/is convicted of committing a crime, which is too broad IMHO.
Depending on the illegal business in question, there may be more contextual suggestions...
is someone who commits crimes involving business activities, mostly desk-work crimes like accounting or insurance fraud.
Assuming "to do illegal business" means "to be in control of a criminal organization", it's crime boss or crime lord.