Tech entrepreneur Andrew Yang raised $10 million in the past three months, his campaign said Wednesday—greatly outpacing at least two senators in the Democratic presidential race and showing that his offer of free money to supporters is paying dividends.

At the debate last month, Mr. Yang—a newcomer to politics—offered to spread $120,000 in campaign money among 10 families in what he described as a pilot program for his chief policy idea. He wants the government to pay every adult American a universal basic income of $1,000...