Veterans Visa and Protection Act of 2021 This bill provides permanent resident status and citizenship eligibility to certain noncitizen veterans and addresses related issues. The Department of Homeland Security (DHS) must establish a program that allows (1) eligible noncitizen veterans outside of the United States to be admitted as permanent residents, and (2) eligible noncitizen veterans in the United States to obtain permanent resident status. Generally, an eligible noncitizen veteran under this bill is one who was not removed from the United States for certain crimes related to violence or national security. DHS may waive these eligibility requirements for humanitarian, family unity, public interest, or exceptional military service reasons. A noncitizen veteran who obtains permanent resident status under this bill shall be eligible for (1) naturalization through service in the Armed Forces, and (2) military and veterans benefits as if they had not been removed (or ordered removed) from the United States. For a noncitizen veteran in removal proceedings or subject to a final removal order, the Department of Justice (DOJ) must determine whether the veteran is eligible for permanent resident status under this bill. If the veteran is eligible, DOJ must adjust the veteran's status. The bill also protects noncitizen veterans or service members from being removed from the United States unless the veteran has been convicted of a crime of violence.