Veterans Economic Recovery Act of 2021 This bill addresses the employment and retraining of veterans, specifically during the COVID-19 (i.e., coronavirus disease 2019) public health emergency. The bill requires the Department of Veterans Affairs (VA) to implement a program under which the VA must provide up to 12 months of nontransferable retraining assistance (including a housing stipend) to up to 35,000 eligible veterans for the pursuit of a specified program of education. Under the bill, an eligible veteran is an individual who, among other requirements, is unemployed due to COVID-19. Veterans who receive retraining assistance may only use such assistance to pursue specified programs, including those designed to provide training for high-demand occupations. The bill requires the Department of Labor to conduct outreach to veterans participating in the program to notify them of employment placement services and facilitate employment placement. Additionally, the VA must enter into a memorandum of understanding with one or more qualified nonprofit organizations to facilitate the employment of veterans participating in the program. The Government Accountability Office must report on the outcomes and effectiveness of the retraining assistance program. The program will terminate 21 months after the date of the enactment of this bill.