The "wains," of course, are waggons as you say -- in fact, "wain" is
etymologically the same word as "waggon." At any rate, Tolkien did not
mean "wains" as "chariots": from App. A, I (iv), "they journeyed in great
wains, and their chieftains fought in chariots." I.e. the "wains" are
different from the "chariots."
Given the geography of Middle-earth as, roughly, Europe,
when one hears of people moving from the East and attempting to settle in the
West of Middle-earth, one thinks foremost of the many nomadic tribes which
attempted to move westwards off the Eurasian steppes into Europe (Avars,
Magyars, Mongols, etc.) or of other tribes who dwelt in the steppes (Scythians,
Cimmerians, etc.).
On the other hand, while these peoples constantly fought
from horseback, to my knowledge they never used chariots. I gladly accept
correction, of course. Yes, Herodotus does mention that the Scythians
used waggons and carts; but he never says anything about chariots and describes
their army always as Cavalry.
War-Chariots are best known from the Bronze Age Aegean and
Near East (i.e. before 1200). In Bronze Age armies the Chariotry
corresponded roughly to the Cavalry of later periods: Mycenaeans,
Hittites, Egyptians, and others in the Bronze Age all maintained a Chariotry as
their armies' élite unit. In the Near East at least, War-Chariots were
used well down into the Iron Age (Judges, IV-V, speaks of Sisera's dread
chariots of iron with regard to a battle probably in the 12th century B.C.;
Shalmaneser III in the 8th century B.C. mentions Chariotry in his Palestinian
opponents' armies, albeit alongside of Cavalry).
It may be that Tolkien here was thinking of no specific
historical people, but was conflating various details for his fictive
Wainriders. I.e. instead of a Cavalry, they use a Chariotry as their
élite arm -- like the Bronze Age peoples of the Levant; but they journey in
waggons and carts like, e.g., the Scythians or even the so-called Sea Peoples
who attempted to invade Egypt in the early 12th century B.C.: despite their
name, they are depicted in Egyptian reliefs with their wives and children
travelling in ox-drawn waggons.
So, the Wainriders are like e.g. the Magyars or the Avars
-- only they aren't (yet?) using Cavalry. Instead, they're using a
Chariotry, historically speaking, a more primitive type of armed unit.
Anyway, it's late; and I'm tired; and that's the best that
I can do.
;-)

A master of disguise, the fox easily
blended in with Faramir's rangers when
Frodo and Sam were captured.
(Errol Flynn, eat your heart out:
you're not even in the same league...)