Each West Yorkshire authority sets its own permit fee and notice period for skips placed on the highway, so ask the supplier how they handle the permit for your address. A skip on your own drive or private land needs no permit.
For a specific town see Leeds, Bradford, Huddersfield, Halifax or Wakefield. For anywhere else, search your postcode to see who covers it. The size guide works out the size from what you are clearing and how heavy it is.