Both my spouse and I are members of the Hitachi Health Insurance Society as employees (insured persons) and our child is the husband’s dependent. All three of us have been vaccinated. How should we apply?

If both spouses are insured persons (i.e., they have different health insurance codes and numbers), each spouse should apply for his or her own vaccination individually.
The insured person on whom the family member depends (in the case of this question, the husband) should apply for the dependent's vaccination.