Well, it's the responsibility of the former owner to fill in Part B of the vehicle registration document and send it to the Driver And Vehicle Computer Services Division in Shannon, who then send a new registration document out to you. If you still have that person's contact details, give them a ring and ask them to do it. 

 

If you can't get them, and you have proof of sale or a receipt of some kind, then get in touch with the Driver and Vehicle office in Shannon (tel: tel: 0818 20411 20412) and request a replacement log book (it costs €12).