As far as I know the Poetic Edda does not mention Thor's vehicle at all. It does mention in Hymiskviða, where one of his goats is famously rendered lame, that his conveyance is pulled by goats. This is also mentioned in the poem Þrymskviða which mentions them 'hurrying in shackles' (skyndir at skǫklum).

As with most things the more direct description is given in Snorri Sturluson's Edda (AKA the Prose Edda), which says in Gylfaginning (Guðni Jónsson's edition):

Þórr á hafra tvá, er svá heita: Tanngnjóstr ok Tanngrisnir, ok reið þá, er hann ekr, en hafrarnir draga reiðna. Því er hann kallaðr Öku-Þórr.

Meaning [my rather-literal translation]:

Thor owns two billy-goats, who are named thusly: Tanngnjóstr and Tanngrisnir , and a ride which he drives which is pulled by the billy-goats. Therefore he is called "Rolling Thunder".

Thor's name means "thunder" ("Thunder" in English has the same root too, for that matter). So the term here Öku-Þórr here which means "rolling thunder" also can be read as "driving Thor". The modern word for thunder in Danish/Norwegian torden is 'Thor noise', and the modern Swedish term is the euphemistic åska ('driving [pagan] god').

Now I translated the noun reið here as 'ride'. Because that's the primary literal meaning of the term "a ride, riding, a horseback journey". Snorri Sturluson seems to have used the term figuratively to mean 'cart, wagon' much in the way "ride" is used figuratively in English for vehicles. Using the term in this sense was not super-common (Snorri accounts for many of the attestations), but for instance Knýtlinga saga uses lǫgreiða (sea-ride) as a poetic term for ship.

The reason why Snorri used this particular term is fairly obvious though, because the term, most commonly in its plural form (reiðar) is also a word meaning 'thunder, rain, thunderstorm', which no doubt comes from Thor's connection to the weather. So from a literary perspective it's sensible he used the word reið for 'vehicle' that connected to his weather role, rather than a more common term like vagn (wagon) or kerra (cart). However this also means it's a an even more ambiguous term.

It's hard to escape the most direct interpretation of reið when used as a word for bad weather is that it comes from a notion Thor was riding. Besides being the most common sense of the word, it makes sense that a thunderstorm would be 'a ride' in the sense of a travel of Thor's across the sky. "A ride" in the sense of his vehicle is.. less obvious. Unlike English 'ride', the term reið when used for a journey only refers to ones on horseback, it's not been attested for figuratively referring to wagon-rides. At the same time the verb used to describe Thor's traveling, aka means 'to drive'. So we have two terms here "reið(ar)" ('ride(s)') for thunder/rain storm and "Öku-Þórr" ("driving thor") that differ in the implied mode of transport. It's contradictory. Yet the term reið for the weather is well attested, but so is Thor having some goat-drawn vehicle. Snorri apparently reconciled it by using "reið" as the term for his vehicle, even though it's still not a great fit for how reið-as-in-storm was used.

As for actual two-wheeled chariots, they seem to have existed in Bronze Age Scandinavia, in the second millenium BCE. One is clearly seen for instance on the rock art of the Kivik grave. This is a good 2,000 years before the Viking Age though (twice as far removed from it as we are) and chariots are not known to have been used by the Viking Age. Old Norse did not have a specific word for them. Even the modern languages don't, the term used is "battle wagon". (or "horse battle wagon" in Swedish and Norwegian where the unqualified term has come to be used for tanks)

The carts we have from the Viking Age were those that were buried ceremonially. (which is just as well, as a religiously-significant cart is perhaps a better idea of what people would've imagined Thor had than an everyday cart). The best known and most ornate example of which was the one from Oseberg in Norway (circa 830), although simpler ones have been found in Lindholm-Høje in Denmark. So that's what a fancy, high-status wagon looked like at the time. To some extent seems there was some convention of burying high-status women in carts in a possible analogue to ship burials. (To be clear: ship burials didn't occur by pushing burning boats out to sea. The ships were burned and then buried on land, or just buried.)

So Thor's "chariot" is just not there in the original text. It is entirely a product of later history. For instance Mårten Eskil Winge's painting from 1872 of Thor battling giants depicts Thor in a chariot. A larger cart would perhaps not work so well with the composition. Perhaps the connection to classical antiquity made chariot seem like a more suitable conveyance for a god. In any case it's almost certainly the most celebrated painting of Thor, one of the more popular works at Sweden's National Museum, and no doubt influential on later depictions of Thor.

On an aside, other details of he painting more informed; his hammer is clearly based off its form in Viking Age pendants. A swastika is seen on Thor's belt, which appears informed by the recent (1867) been suggestion of runologist George Stephens that it was a symbolof Thor, based on its occurrence on the Thurmuth sword. This theory didn't gain much support though. (And in terms of swastika-history, it would soon be eclipsed by Schliemann publishing the ones he'd found in Troy and the suggestion it was an old 'Aryan' symbol)

Anyway, the short of it is that the texts don't say anything as specific as 'chariot' as distinct from wagons/carts, and without going out of their way couldn't describe a chariot in a single word, as there was no specific term. Kerra is the term used most commonly (although not exclusively) for two-wheeled carts and was the term used when for instance Old Testament chariots were rendered into Old Norse in Stjórn. (Pharaoh let's Joseph ride in sealfs sins kerru 'his own chariot' in the paraphrase of Genesis 41:43)

But as said there's nothing of even that specificity. Even a sled could be a possible reading if it hadn't been for the fact that the rumbling/rolling sound of thunder is obviously central to the myth