Build Housing with Care Act of 2023 This bill establishes a competitive grant program to support co-located childcare and housing facilities. It also requires the Government Accountability Office to study the availability and affordability of childcare for residents of public housing. Specifically, the Department of Housing and Urban Development must award the grants to eligible entities, which include community development financial institutions, public housing authorities, licensed childcare providers, government entities, housing developers, and consortia of such entities. Entities must use the grants for designing, planning, constructing, acquiring, renovating, or otherwise supporting new or existing housing facilities with childcare providers on the same premises or nearby.