Reform and Update Rural Access to Local Exams Act of 2021 or the RURAL Exams Act of 2021 This bill updates policies and procedures related to covered medical disability examinations provided by the Department of Veterans Affairs (VA) and its contractors. Under the bill, covered medical disability examinations are those that are required for purposes of adjudicating specified VA benefits. First, the bill requires the VA to begin collecting data to improve veterans' access to covered medical disability examinations. The VA must study rural veterans' access to covered medical disability examinations and submit a plan to improve access. Additionally, the VA must ensure that contracts for the provision of covered medical disability examinations by contractors include specified elements. Specifically, such contracts must (1) include financial incentives to encourage timeliness in medical disability examination appointments for rural or housebound veterans, (2) include financial disincentives to encourage timeliness, and (3) require contractors to cooperate with inspections. Finally, the VA must establish a program of periodic inspections of sites, locations, and facilities where medical disability examinations are provided by VA contractors.