John Menard Jr. is widely known as the richest man in Wisconsin. A tough-minded, staunchly conservative 75-year-old billionaire, he owns a highly profitable chain of hardware stores throughout the Midwest. He’s also famously publicity-shy — rarely speaking in public or giving interviews. So a little more than three years ago, when Menard wanted to back Wisconsin Gov. Scott Walker — and help advance his pro-business agenda — he found the perfect way to do so without attracting any attention: He wrote more than $1.5 million in checks to a pro-Walker political advocacy group that pledged to keep its donors secret,...