I think of the battle chariot as a weapons platform of the Bronze/Iron Age. And according to Wikipedia, the saddle-hung stirrup wasn't popularized in Europe around the 6th century.
Were battle chariots still used in the Middle Ages? If not, was it perhaps because the stirrup took its place as a better means for stabilizing horse-powered archers/lancers?
These two matters are not related. As you say, there is more than a millennium between the time the chariot went out of use and the time the stirrup was introduced.
The chariot was made obsolete by the rise of "true cavalry" - organised units of men fighting on horseback. These were more mobile and versatile than chariots, as well as being more numerous, on account of being cheaper. The development of superior horsemanship and the breeding of stronger horses allowed for a more effective horse-based military force than chariots. The first people to make extensive use of true cavalry were the Assyrians at the time of the Neo-Assyrian Empire (10th-7th century BC). They were known to field very effective cavalry, as were the Persians, Greeks, Macedonians and Parthians of later times - all before the appearance of the stirrup in the West.