B737 Captain reported a horizontal stabilizer trim malfunction during initial climb. The crew hand flew the aircraft with manual trim inputs to the destination and landed.

Narrative

Shortly after rotation on takeoff from XXR ZZZ; I tried to apply a small amount of trim from the yolk switch and got no response. My FO then tried his yolk switch with same result. I continued the NADP (Noise Abatement Departure Procedure) departure using the crank handle on the trim wheel. Once the airplane was clean configured and climbing I tried to turn on the A/P (Autopilot) while about the same time as we got a warning annunciation FLT CTL". About this time we also got a STAB OUT OF TRIM light. I continued to hand fly and handle ATC comm; while He worked the checklists in the QRH. After completing the procedures in the FM (Flight Manual); he contacted Dispatch and had us patched into Maintenance Control with instructions to include the Chief Pilot. After consultation it was agreed then we had more than enough fuel to continue to ZZZ1 below RVSM altitude. We agreed that the conditions and weather along the route and at destination were good enough to allow us to hand fly the aircraft without difficulty. (The FM procedures had us with no AP; AT (autothrottles); or electric trim; and the requirement to land Flaps 15.). The rest of the flight we took turns hand flying in 30 minute shifts. The subsequent approach and landing at ZZZ1; were as expected and uneventful. We landed with over 2 hours of fuel."