The earliest iteration of the Táin is dated to the 9th century AD (though the content itself may be earlier than the time it was committed to writing) in the Early Medieval in Ireland (roughly AD 400-1000). There is a lot of debate over whether the story and the contents are a wholly native tale or a work which was composed with heavy reliance on biblical and wider European tropes in storytelling (which in no way diminishes its importance). Apart from the obviously fantastical elements like Cúchulainn having 7 pupils in each eye there have been people who suggest that the Táin provides us with what many people called 'A window on the Iron Age'. The idea that a story first written down in the 800's AD could provide somewhat accurate information about pre-Christian (pre 400 AD) life in Ireland is also quite heavily debated though until quite recently it was a fairly uncontroversial idea.

So, now that I've gone into why we might want to take the Táin as evidence of activities which took place in Ireland's past with a pinch of salt, I would like to point out that there is no archaeological evidence in Ireland that I know of that indicates a vehicle like the chariots depicted in the Táin existed in the Irish Iron Age (800 BC - 400 AD) or the Irish Early Medieval Period.

With that said, we do have evidence from elsewhere for the existence of chariots which might have been the type of thing the composers of the Táin had in mind when they wrote down (or made up) the events of the story. In terms of chariot warfare Julius Caesar describes British peoples using them rather extensively during his ultimately failed invasion of Britain in 55 BC. The chariots Caesar faced were two-man vehicles with a driver and a warrior which harassed the Romans by driving across the front line of the enemy at a distance where missiles such as javelins and darts could be thrown in an effort to break up the force's cohesion. The warriors would jump down from the chariot and engage with spears, swords, and shields with the chariot acting as a pickup service in case the warrior got into a spot of bother. Caesar describes them as working closely with cavalry, specifically that the chariots would retreat between cavalry units as a protection against enemy cavalry. What he also mentions is the limitations of chariots: they need a lot of room to maneuver as well as good even terrain. There are plenty of examples of these types of chariots in high-status burials in Britain, particularly in Yorkshire, such as the Wetwang Slack burial. The fact that chariots were buried with these evidently wealthy and important people makes it likely that they were as much a status symbol as a functional piece of martial equipment.

While Ireland has plenty in common with Britain during this period of history (the Late Iron Age) I do not think we can reasonably assume the existence of chariot warfare of the type described by either Caesar or in the Táin without hitherto unknown archaeologial evidence coming to light.

Sources:

Caesar. The Gallic War. Translated by: Edwards, H. J. 1966.

Gosling, P. 2011. Placing Names in Táin Bó Cúailnge: The river 'Níth' and the ford 'Áth Carpat'. Journal of the County Louth Archaeological and Historical Society. Vol. 27(3). Pp. 366-378