Wheeled transport vehicles were available only for the Imperial family and the highest aristocrats, but these were even slower than the palanquins: called goshoguruma (gosho refers to the Imperial palace), these were ox drawn carts, but since there was no driver seat, the driver was walking along the oxen...
Here is a representation of such goshoguruma, drawn on a folding screen from the Heian Period, a moment from the famous Tale of Genji displayed at the Kansai Airport.
EXIF info:
Nikon D90
Lens: VR 18-55mm F/3.5-5.6G
Focal Length: 18mm
Aperture: F/4
Shutter Speed: 1/30s
ISO Sensitivity: ISO 500