Daniel Webster Congressional Clerkship Act of 2018 This bill establishes the Daniel Webster Congressional Clerkship Program for the appointment of individuals who are graduates of accredited law schools to serve as congressional clerks in the Senate or House of Representatives. The Senate Committee on Rules and Administration and the House Committee on House Administration must each select at least six individuals for a one-year term to serve for a clerkship with an office in their respective chambers. The committees shall ensure that congressional clerks selected under this bill are apportioned equally between majority and minority party offices.