Has there ever been cases of someone with a successful legitimate business paying the taxes using illegal money?

If they're paying business taxes with illegal money, it isn't a legitimate business. It's one that's laundering money.

Sort of. John DeLorean, who started the company that made the cars with his name on them (the Back to the Future car), and the guy who designed the Pontiac GTO, Firebird and Grand Prix, was lured into a deal to sell cocaine to raise cash for his failing car business. He never actually sold any drugs, because the FBI was to supply the drugs. A career criminal who knew DeLorean causually and knew of his business troubles saw a way to get himself a sweeter deal on his own criminal charges and went to the FBI as an informant, lured DeLorean into a trap, and DeLorean was arrested. He was found not guilty on the grounds of police entrapment - he was targeted because he was financially vulnerable. But it was the end of his business career.

Yes. A business has the opportunity to engage in crime and keep it secret. The crime is punishable regardless whether profits are used for business expenses or taxes.