Yes. L-2 dependent children may enrol in school. They may not work, and their dependent status ends when they turn twenty-one.
Study is allowed; employment and the age limit are not negotiable
Hypothetical example: Dependent children admitted in L-2 status are among the nonimmigrant classes permitted to study, so enrolment in public or private school is straightforward and needs no separate permission. Two limits deserve attention in advance. First, an L-2 child is not employment-authorised, which matters where a teenager would ordinarily take summer work; the L-2 spouse's employment authorisation does not extend to children.
Second, dependent status ends at twenty-one. A child who arrives at nineteen for a five-year L-1B assignment will need an independent basis to remain, most often student status, well before the birthday rather than after it. For a family planning a move around a school year, put both children's twenty-first birthdays on the same page as the assignment end date, and check whether the plan still works when those dates are read together.