Yes. Children in L-2 status may attend school for the duration of their dependant status, without separate student status, but they may not work.
School is permitted; employment and the 21st birthday are the limits
Hypothetical example: A Portugal Cove-St. Philip's developer's spouse and unmarried children under 21 may accompany in L-2 status. Children may enrol in public or private school as dependants; the ICE list of nonimmigrant classes that may study confirms that L-2 dependants can attend school without changing to a student status. The two restrictions to plan around are that L-2 children have no employment authorization, so a teenager cannot take a part-time job at the hatchery or anywhere else, and that dependant status ends at 21. If a child will reach 21 during the assignment, that child needs a separate basis to remain, and it should be arranged before the birthday rather than after. The L-2 spouse, by contrast, is employment-authorized incident to status. A dependant's permitted stay is tied to the principal's, so a school year that extends beyond the L-1B validity requires an extension of the principal's status first.
Hypothetical example: a vertical-farming automation engineer relocates with a spouse who has been offered a teaching position and a child nearing university age. The family should line up each person's expected activity with the status that will be requested, then identify documents needed at admission. The spouse should retain evidence of the appropriate dependant classification before accepting paid work, while the child needs a separate plan for the point at which dependent eligibility ends. A calendar showing admission, school enrollment, and the child's birthday makes the decision concrete rather than leaving it to a later emergency.