The word consignment means that you are giving control of something, or in this case someone, to another. In the way it is used in this text, it is referring to sending someone to prison.
Mr. Lorry speaks of ". . . the privilege of filling up blank forms for the consignment of any one to the oblivion of a prison for any length of time. . ." meaning the right to send anyone you want to jail for however long you choose.