Ms. Abedin’s high profile and proximity to Mrs. Clinton also brought her scrutiny. Senator Charles E. Grassley, Republican of Iowa and chairman of the Judiciary Committee, has questioned Ms. Abedin’s arrangement to earn income privately while she worked for Mrs. Clinton at the State Department. In addition to being on Mrs. Clinton’s personal payroll, Ms. Abedin received money from the Clinton Foundation and Teneo, a consulting firm co-founded by Douglas J. Band, a former senior aide to Mr. Clinton. And some of Ms. Abedin’s emails on Mrs. Clinton’s private server led to accusations that foundation donors had received special access to the State Department.