“chaqu’un” doesn’t exists at all in French language. The unique construct is “chacun” (or “chacune” for the female form), translated to “each one”.
The word came from ancient French “chascun”, which is derived from the Latin “cascunum”.

chaqu’un is not a real word and must not be used.

chacun seems to be originating from a contraction of latin roots of chaque un and un à un. (wiktionary.org)

Like everyone else said chaqu’un doesn’t exist.

Chacun comes from old French “chascun” which in turn comes from the Vulgar Latin word cascúnum (a crossing of “quisque(unus)”, that literally means “every one” = « chaque un ») and catúnum (from [unum] cata¹ unum), literally « un à un »).

You can read more about it on the CNRTL site definition for “chacun“.

^{1: It comes from Greek κατά “exprimant une notion de temps, avec idée de distribution: κατἐνιαυτόυ « chaque année »”}

The word is chacun. The word chaque is always spelled with the final e anyway. I’m surprised to find so many Google hits for chaqu'un; I wouldn’t have thought it to be a common spelling error.

Etymologically speaking, chacun is related to chaque, but it is not a contraction of chaqu’un. It’s rather the opposite, in fact. The word chacun comes from the vulgar Latin word cascunum (with a similar meaning), which in turn seems to be a cross between quisque unus (“each one”) and unum cata unum (“one by one”). The contraction into one word happened back in Latin, not in French. The word chaque arose later, in Old French, as a derivative of chascun (as it was then spelled), by analogy with quelque and quelqu’un.

If you look at published books, you’ll see that there is no contest: the spelling is chacun.

The only occurrences of chaqu’un are in older texts (early 19th century is the most recent I could find), and there are very few of these. So I would say that it’s an old spelling error, but a spelling error nonetheless.

Looking at some old dictionaries, I find no mention of a recognized variant spelling. Féraud’s dictionary does list chaqun as an incorrect spelling for chacun.

Looking at modern usage, the Leipzig corpus (which includes many newspapers and websites such as Wikipedia) has very few occurrences of chaqu. The usage examples are striking by the number of other, uncontroversial spelling errors they contain. So I feel confident in dismissing chaqu’un as incorrect, even under the most descriptivist approach.