Columbus Nova, an investment firm with ties to a Russian oligarch, burst into the public eye last week when it was revealed to have paid $500,000 to Michael Cohen, President Donald Trump’s personal lawyer.

The firm, founded in 2000, manages the funds through which Viktor Vekselberg, one of Russia’s richest men and a target of recent U.S. sanctions, has invested in American companies, real estate and other opportunities. The company, run by a cousin of Mr. Vekselberg, says the oligarch doesn’t control it but has provided most...