N.H. Rev. Stat. Ann. § 671:28-a

If a school board member is absent from or unable to perform his or her duties for all or any part of the day at any school district election, the school board member is authorized to appoint a school board member pro tem to perform his or her duties. If one or more school board members are absent or unable to perform their duties and have not appointed school board members pro tem, or if a school board member pro tem is absent or unable to perform his or her duties for all or any part of the day, the moderator is authorized to appoint school board members pro tem to perform their duties.