Veterans' Appeals Backlog Improvement Act This bill requires the Department of Veterans Affairs (VA) to establish programs that provide employment opportunities within the VA and the Board of Veterans' Appeals (BVA). Specifically, the bill requires the VA to establish a competitive internship program to provide high-achieving law students with an opportunity to gain experience with the BVA. The VA must also establish a nine-year competitive honors pilot program to recruit high-achieving law students, recent law school graduates, and entry-level attorneys for employment. Under the pilot, the VA must (1) provide student loan repayment benefits to participants, (2) enter into participant agreements that outline terms of the program, (3) assign mentors who are outside of the participant's chain of command, and (4) provide participants with at least one assignment within the VA's Office of General Counsel. The bill also requires the VA to report on improving access to tele-hearings before the BVA (held by picture and voice transmission).