Morgan Snipes downloaded the Robinhood Markets Inc. app on his phone nine months ago with a plan to leave his day job behind.

The stock market has turned into a game for inexperienced investors like 35-year-old Mr. Snipes, a small-business owner from Oklahoma City, thanks to commission-free trading apps. With simple designs, alerts and the ability to win free shares for recruiting friends, apps such as Robinhood, Webull, M1 Finance and JPMorgan Chase & Co.’s You Invest have drawn millions of users who are typically young,...