B737-800 Flight crew reported a near miss with a moving vehicle on the ramp.

Narrative

After landing; we were taxiing into gate X from spot XX. We stopped short of the gate due to a baggage cart inside the safety area. We had turned facing the gate but short of parking. Brakes were set; and after the cart was clear and the marshaller and wing walkers gave the signal to proceed; I released brakes; pushed up power and starting moving into the spot. We had only moved a few feet when out of my peripheral vision I saw the left wing walker waving his wand and starting to jump/run towards the aircraft. I looked left and saw a blue truck driving underneath my wing from the rear of the airplane. I immediately and abruptly stopped the airplane. The truck was now in front of the wing and very very close to the aircraft. Once the wing walker had caught the attention of the driver; he turned and drove away. We continued to park uneventfully. The wingwalker came to the flight deck after we shut down and said the truck was also very close to our left engine; which was running at the time.Cause: This event was caused by a poorly trained or inattentive driver on the ramp. The wingwalker did not see the vehicle sooner; and perhaps did not clear the area well before we were given signals to proceed. The spot where we were stopped short of the gate was also over vehicle road markings from the north where it intersects with the vehicle east/west road along the gates. It appears the driver was on the road" even though it went under my wing."