Migrant Child Safety Act This bill requires actions relating to placing unaccompanied alien children with a sponsor, such as requiring the Department of Health and Human Services (HHS) to collect and share specified information about prospective sponsors. Before placing such a child with a prospective sponsor, HHS must gather and provide to the Department of Homeland Security (DHS) specified information about the prospective sponsor, including (1) proof of address for the residence where the child will be placed; (2) results of background checks for the sponsor and any adult household members, (3) the sponsor's immigration status, if known; and (4) any evidence to prove any claimed relationship between the child and the sponsor. HHS may not waive any part of this requirement. After receiving this information, if DHS determines that the sponsor is unlawfully present in the United States, the sponsor must be placed in removal proceedings. No later than 30 days after placing the child, HHS must also provide this information and any other relevant information to the appropriate state or local health or welfare agency. If HHS cannot contact the sponsor for a follow-up or well-being check during the 120 days after the child's placement, HHS must report the child as missing to the National Center for Missing and Exploited Children. HHS must attempt to contact the child's parents, to the extent practicable.