Honoring the Oath Act of 2021 This bill establishes the Military Family Immigration Advisory Committee and modifies the naturalization procedures for a noncitizen in the Armed Forces. U.S. Immigration and Customs Enforcement shall notify the committee when any individual in removal proceedings is a (1) member of the Armed Forces, (2) veteran, or (3) noncitizen spouse or minor child of such an Armed Forces member or veteran. The committee shall review the case of any such individual and make recommendations to the Department of Homeland Security (DHS) and the Department of Justice as to how the case should be resolved. Such an individual shall not be removed until the committee has provided its recommendations. DHS and the Department of Defense shall jointly carry out a program allowing a noncitizen individual who serves on active duty in the Armed Forces (and any noncitizen spouse or minor children of the individual) to become a U.S. citizen if the individual is not otherwise ineligible for citizenship. The Armed Forces member shall be given the opportunity to apply for citizenship during the accession process into the Armed Forces. DHS shall adjudicate such an individual's application by the last day of the individual's active service in the Armed Forces.