B737-800 Captain reported Nose Wheel Shimmy during landing roll out accompanied by lack of Nose Wheel Steering. The flight crew requested a tug to move the aircraft to the gate. Maintenance reported the Nose Wheel Scissors Bolt was missing.

Narrative

FO Landing Runway XX ILS ZZZ; Flaps 30; Brakes 3; Gross Weight 133;000 lbs.; Wind 010 at 12 Gusting 20; VMC conditions on the approach. After Normal easy landing in the touchdown zone at the 1;000 ft. marker; normal auto braking applied; spoilers deployed normally; and auto brakes came off when manual braking was applied. Slowing through 80-60 kts.; transferred aircraft control to the me; the Captain. At this point is when the aircraft started to shutter heavily like maybe we had a blown nose tire. I continued down Runway XX slowing the plane to less than 20 kts. while the shaking dissipated. We made a right turn off at Taxiway XY using differential breaking as there was no nose-wheel steering capability and cleared the runway; stopping short of Taxiway XX On Taxiway XY. We informed Tower we may have blown a tire; and needed to be inspected. We made a PA to the passengers that we may have had a tire failure and would need to be towed to the gate once Maintenance inspected us. We called Company Operations and requested Maintenance to inspect us and coordinated for a tug. 2 Port Authority trucks inspected us and informed us that we had a part/piece hanging down behind the nose wheel. APU was started; we shut the engines down; Company Maintenance arrived and informed us through inter-phone that our nose scissors bolt had came out and the scissors disconnected/came apart. Company Maintenance reconnected the scissor arms and towed us to Gate XX for a normal gate arrival and passengers; deplaning normally through the jet-bridge. The process from after landing to the gate took 40 minutes. We did not disrupt any traffic landing or taking off from ZZZ. A Logbook entry was made at the gate for nose scissors bolt missing causing linkage to separate and the aircraft was taken out of service by Company Maintenance. Push back; taxi; takeoff all normal flying out of ZZZ1. There was nothing unusual or abnormal about the landing. It was nice soft landing and the nose wheel was lowered at a normal rate and nothing rough or abrupt about it. Aircraft control was maintained throughout the rollout at ZZZ and remained either on the center line of the runway or the taxiway lines the entire time of the event. Operations and Company Maintenance worked well together to get the plane to the gate and the passengers safely to the terminal.