Veteran Service Recognition Act of 2022 This bill addresses immigration-related issues pertaining to noncitizen (or non-U.S. national) military veterans, including by authorizing the Department of Homeland Security (DHS) to provide lawful permanent resident status to a veteran subject to removal. If a noncitizen veteran appears to be eligible for lawful permanent resident status under this bill, that veteran must receive a reasonable opportunity to apply for such status and may not be removed until there is a final administrative decision on the veteran's eligibility. For the purposes of providing such status under this bill, DHS may waive any applicable grounds of inadmissibility, except for certain crime- or security-related grounds. The bill also extends certain deadlines relating to obtaining citizenship after serving in the Armed Forces. DHS must create a system for identifying noncitizens who are or may be veterans. Before initiating removal proceedings against a noncitizen, DHS must attempt to determine whether the noncitizen is a veteran. DHS must ensure that veteran status is considered in immigration proceedings. The bill also requires U.S. Citizenship and Immigration Services (USCIS) and the Department of Defense to take certain actions to facilitate opportunities for military recruits and veterans to obtain U.S. citizenship, such as by training Judge Advocate General Corps members to act as liaisons between the Armed Forces and USCIS on servicemember citizenship applications. The bill also establishes the Military Family Immigration Advisory Committee to provide recommendations on cases involving the removal of a servicemember, veteran, or certain family members of such an individual.