A federal judge in Missouri has denied a Competitor Group Inc. request to dismiss a lawsuit charging that the race operator’s use of volunteers violates federal labor laws.

Like nearly all such races, CGI’s Rock ‘n’ Roll series of marathons and half marathons rely on volunteers to staff water stations, hand out finisher medals, dispense t-shirts and more. Yet unlike many race operators, CGI operates for profit as a subsidiary of Calera Capital, a private-equity firm. Last September, a St. Louis University law professor named...