No. Children in L-2 status may attend school but have no employment authorization at any age; only the L-2 spouse is authorized to work incident to status.
School yes, employment no, and a birthday to watch
L-2 dependant status for the children of a Torbay planner permits residence and study; the ICE list of nonimmigrant classes that may study confirms that L-2 children can attend school without separate student status. It does not permit employment, and there is no age at which that changes short of the child obtaining a status of their own. A sixteen-year-old cannot take a summer job at a store, and a nineteen-year-old cannot work a season at the mine, whatever the local employer says about paperwork. The L-2 spouse is in a different position and is employment-authorized incident to status. The other date to watch is the 21st birthday: L-2 status for a child ends on that day regardless of the planner's remaining L-1B time, and a child who will turn 21 during the assignment needs an independent basis to remain, arranged before the birthday. Dependants' status also ends when the principal's does, so extensions must be filed in time.
Hypothetical example: the sixteen-year-old child of a Torbay ceramic-kiln diagnostics specialist is offered weekend work by a U.S. retailer. The family should decline until the child obtains independent work-authorized status; school enrollment does not create that permission. The first review should place the child's birth date, the principal's remaining L-1B time and any future education plan on one calendar. That record reveals whether the family needs a separate status strategy before the child turns 21.