Veteran Readiness and Employment National Emergency Extended Deadline Act of 2021 or the VR&E NEED Act of 2021 This bill extends a veteran's eligibility for 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 a veteran from participating. Specifically, the period of eligibility must not run during the period the veteran is prevented from participating in the program. The period must resume on a date, determined by the Department of Veterans Affairs, that is not earlier than the first day after the veteran is able to resume participation and not later than 90 days after that day.