In the United States this would be a very difficult question to answer, as land use law not only varies from state to state, but from city to city within the state. Many different departments would be involved, especially in a major project in a big city. In our little city of thirty thousand, these departments might be involved: a) the building inspector, which grants building permits b) the planning board, which decides what kinds of things can be built and sited where and creates the zoning map c) the zoning board of appeal, which has the power to grant zoning exceptions if you can justify them d) the conservation commission, which has a say on preserving open space and preventing the destruction of wetlands

This will vary from place to place. Where I live, we would say 'the planning department'.

In general, that would be called a 'Town Planning Department', or just 'Planning Department'. Here - it is part of the City Council.