I was looking at overland transportation in ASOIAF. A lot of lords and ladies just use horses, but sometimes vehicles are used.
Wheelhouses - Cersei obviously has a really huge double-decker royal wheelhouse, drawn by 40 draft horses. The Tyrells have a wheelhouse adorned with gilded roses carved into it. From my understanding, it seems like wheelhouses tend to be for longer journeys, but a royal wheelhouse transported Rhaella the short distance from the Red Keep to a waiting ship after the loss at the Trident
Palanquins - seem to be more of an Essosi thing, but Tyrion knows what they are, so they may exist in Westeros. Can be open-air or enclosed by light curtains / heavy drapes. Carried by slaves, later by freed men. The windblown have a commander who has a palanquin with 40 bearers to carry it.
Sedan chairs - again, these seem to be more of an Essosi thing, as they’re only mentioned in Dany’s chapters. More open than a palanquin, and you sit upright.
Carriages - mentioned a few times in Westeros. Note that a wheelhouse is described as a type of carriage. Probably very difficult to use on medieval roads, bumpy and uncomfortable.
Litters - A lot of the nobility tend to be carried around King’s Landing in horse litters. For example, Sansa, Jeyne, and Septa Mordane ride together in a large litter with diaphonous silk curtains, to the Hand’s tourney. After being injured by Brandon Stark, Petyr is sent from Riverrun to the Fingers in a closed litter. Tyrion also travels by litter in King’s Landing, as does the High Septon. Doran Martell always travels by litter because of his gout, but it’s considered too cumbersome to travel through mountains.
Litters tended to shield the occupant(s) from the rain or sun.
The litter probably swayed a great deal and, although made more comfortable with cushions, furs and rugs, it would have been awkward to sit in for long distances.The people inside probably suffered from travel sickness, and they would have staggered out stiffly at the end of a journey [x]
In Essos, litters are carried by slaves/freed men or horses. In Westeros, it only mentions horses carrying litters, but they may be carried by people too, idk.
Hathay - a type of ornate cart used in Volantis, pulled by a small elephant and driven by a tattooed slave. It has two large wheels.
There are also oxcarts, wagons, and wayns (a wain, a farm wagon or cart) for peasants and merchants.
Here is an interesting post about travel in the middle ages. If anyone has more to add, or any corrections, please tell me!