Yodlee Inc.’s main business is providing online personal-finance tools to 11 of the 20 largest U.S. banks in assets.

But when bank customers use those tools, Yodlee has another way of making money: The company sells some of the data it gathers from credit- and debit-card transactions to investors and research firms, which mine the information for clues about trends that can move stock prices.

“Yodlee...