GI Bill National Emergency Extended Deadline Act of 2021 This bill amends various provisions related to educational assistance benefits provided by the Department of Veterans Affairs (VA). First, the bill extends the time limitation for using educational assistance under the Montgomery GI Bill and the Post-9/11 GI Bill, and for eligibility under the Veteran Readiness and Employment program, when educational institutions are temporarily or permanently closed due to an emergency situation, or there is another reason that prevents an individual from participating. Next, the bill requires the VA to disapprove courses of education provided by public institutions of higher learning that charge higher tuition and fees than in-state tuition for individuals who are entitled to educational assistance under the Survivors' and Dependents' Educational Assistance program. The bill also makes adjustments to the time limitation for using educational assistance under this program. The VA must implement an information technology service using one or more commercial software systems to process claims for VA educational assistance benefits. Finally, the bill requires the Veterans' Employment and Training Service within the Department of Labor to implement a pilot program under which states may use grants or contracts to carry out a short-term fellowship program to provide veterans with opportunities for long-term employment.