Probably during the First Age
The passage from Unfinished Tales quoted in chepner's answer makes clear that wheeled vehicles, while rare, existed in Númenor "in the early centuries". This puts the invention of the wheel at no later than the early centuries of the Second Age.
In Númenor all journeyed from place to place on horseback;
Therefore the roads of Númenor were for the most part unpaved, made and tended for riding, since coaches and carriages were little used in the earlier centuries, and heavy cargoes were borne by sea. The chief and most ancient road, suitable for wheels, ran from the greatest port, Rómenna in the east, to the royal city of Armenelos ...
Unfinished Tales Part Two, Chapter I: A Description of the Island of Númenor
Page 164 (Houghton Mifflin Harcourt; 2012 Kindle Edition)
I have not been able to find any earlier reference to wheels, but I think we can we can infer that they were invented earlier. The passage quoted from Unfinished Tales makes clear that the Númenóreans had little need for the wheel when they first moved to Númenor so they would have little incentive to invent it.
If the Númenóreans already knew about the wheel when they moved to Númenor, they must have acquired that knowledge in Middle-earth during the First Age. That suggests that the wheel was invented in the First Age. I can find no reference to wheels or wheeled transport in the First Age. So while I think the wheel was likely invented in the First Age, I can't say exactly when or who invented it (although my money would be on the dwarves).
As "the early centuries" is quite vague, I looked for a quote that gave a more more precise date. The best I could find is also in Unfinished Tales; it has a more precise date (between 873 and 892 of the Second Age), but refers to water wheels (presumably for water mills). In our world, the wheel was usually used for transport before water wheels were developed, and it seems reasonable to assume the same applied in Númenor.
All things were made for their service: hills are for quarries, rivers to furnish water or to turn wheels, trees for boards ...
Unfinished Tales Part Two, Chapter II: Aldarion and Erendis
Page 199 (Houghton Mifflin Harcourt; 2012 Kindle Edition)
The quote is from a letter to Ancalimë written by her mother some time before Ancalimë was named heir to the throne in 892 at the age of 19 (so between 873 and 892).
Water wheels were in use in Númenor by some time between 873 and 892 of the Second Age. The invention of the wheel was no later than "the early centuries of the Second Age", and probably some time in the First Age.