Orderly Requirements Designed to Enforce and Regulate Latin American Migration Act or the ORDER Act This bill requires the Department of State to seek to negotiate agreements with Western Hemisphere countries concerning issues related to immigration enforcement and refugee processing. Such agreements must contain certain provisions. For example, the government of Mexico must (1) accept the rapid entrance into Mexico of non-Mexican nationals seeking asylum; (2) process the asylum claims of such individuals inside Mexico in accordance with domestic law and international treaties; (3) accept the continued presence in Mexico of such individuals who have pending applications for asylum in the United States, if the individual is ineligible for asylum in Mexico; and (4) provide appropriate humanitarian protections. Such agreements must also require the governments of Honduras, El Salvador, and Guatemala to accept the entrance of nationals of other countries seeking asylum and process such claims in accordance with domestic law and international treaties.