RIO DE JANEIRO—The contract to hire and train thousands of security screeners for the Olympics in Rio starting Aug. 5 was awarded only two weeks ago to a tiny firm, raising concerns about keeping the Games safe amid global fears of terrorism.

Brazil’s government waited until July 1 to award the contract to recruit and deploy thousands of private security guards needed to monitor X-ray machines and pat down spectators for weapons and other contraband outside Olympic venues, according to documents reviewed by The Wall Street...