Aw, darnit. So much for the Littlest Police Force in New York.
The is no "citizens' arrest" law in New York. The New York Criminal Code allows anyone to use force in defense of themselves or in defense of others. However, such force has to be objectively reasonable in relation to the threat, so if you injure someone who has stolen something of little value, you probably will be charged with assault and sued for causing personal injury. The one exception to this rule is for shopkeepers. The New York General Business Law, Section 218, allows store owners and their employees to stop and physically detain suspected shoplifters until the cops get there.
http://caselaw.lp.findlaw.com/nycodes/c44/a30.html
S 218. Defense of lawful detention. In any action for false arrest,
false imprisonment, unlawful detention, defamation of character,
assault, trespass, or invasion of civil rights, brought by any person by
reason of having been detained on or in the immediate vicinity of the
premises of (a) a retail mercantile establishment for the purpose of
investigation or questioning as to criminal possession of an
anti-security item as defined in section 170.47 of the penal law or as
to the ownership of any merchandise . . . it shall be
a defense to such action that the person was detained in a reasonable
manner and for not more than a reasonable time to permit such
investigation or questioning by a peace officer acting pursuant to his
special duties, police officer or by the owner . . . "