The problem is not so much sneakers as leather; we don't wear leather footwear on Yom Kippur. In the book of Leviticus, the Torah commands us to "afflict" ourselves on this holiest day.
The affliction serves two purposes:
a) On this day, when our connection to G‑d is bared, we are compared to angels, who have no physical needs. b) We afflict ourselves to demonstrate the extent of our regret for our past misdeeds.
The rabbis determined that "affliction" means that we are to deny ourselves of certain "luxuries," among them is wearing leather footwear.
Wearing leather was considered a comfort until recent years when, with the advent of shoemaking technology, it became possible to manufacture comfortable shoes of other materials.
[We also afflict ourselves on Tisha b'Av, when we mourn the destruction of the Holy Temples. Leather footwear is not worn on that day too.]
On a mystical level, wearing leather is reminiscent of the primordial sin committed by Adam and Eve, after which G‑d outfitted them in garments made of skins (Genesis 3:21). On Yom Kippur, the Day of Atonement, we don't want to do anything that recalls this sin. (This is also the reason why we refrain from wearing gold on Yom Kippur, so as not to recall the sin of the Golden Calf.)
As for the "custom" of wearing sneakers, before the advent of plastic shoes and made-in-China replicas, the consumer had to choose between leather shoes or sneakers, so people wore sneakers on Yom Kippur... Today, however, there are synthetic shoes available on the market, and one must be careful when wearing sneakers because many of them are made of leather too.
Click here to visit our comprehensive Yom Kippur section.
Best wishes for a sweet new year,