Moose Hunting is legal in most places but you usually need to get a moose hunting permit from the county or state.
Some good places that moose hunting is legal are Alaska, Idaho, Washington and Maine.
Also New Hampshire is also another good area to Hunt Moose.
Before hunting moose though even in those areas speak with the county or state where you get hunting licenses and ask them what the laws are.
Some places you may need a permit to hunt moose and other places you may not need a permit to hunt moose but it's best to find out directly from the state or location where you plan to hunt the moose.