Liberian Refugee Immigration Fairness Act of 2019 This bill directs the Department of Homeland Security (DHS) to grant permanent residence status to qualifying nationals of Liberia. The program shall be available to a Liberian national who has been continuously present in the United States from November 20, 2014, until the filing of an application for adjustment to permanent residence status, or the spouse, parent, or child of such an individual. The individual must apply within one year of the bill's enactment, be otherwise eligible to receive an immigrant visa, and not be inadmissible under various grounds. Certain grounds for inadmissibility under the Immigration and Nationality Act, such as being a public charge, shall not apply. An individual shall be ineligible if the individual has been convicted of an aggravated felony, convicted of two or more crimes involving moral turpitude, or has participated in the persecution of a person for reasons such as race, nationality, or political opinion. DHS shall promulgate regulations to establish procedures by which an alien who is subject to a final order of deportation, removal, or exclusion may seek a stay of such order based on the filing of an application to adjust status. If there is no such final order, DHS shall stay removal proceedings if a qualified alien submits an application for adjustment of status. DHS shall authorize an individual for employment if that individual's application has been pending for more than 180 days, and may also provide employment authorization before that.