Any day is the more preferred when talking about a general time frame. Usually the term is used to describe something that is happening soon, as in the next couple of days, or any day. This is such an informal term that most dictionaries don’t list it as an actual term on its own.
This term is also used commonly in the phrase any day now, where something is expected to happen very soon but the actual time is unknown. A common example is the time when a woman may go into labor to deliver a child. A due date is announced, but if happening naturally, the exact hour or day is unknown.
Unlike anytime, anyday has not made its way into dictionaries in any form. Argument could be made that if anytime is an adverb, then so is anyday. Like all informal slang, it is up to personal preference in informal works. However, until the time comes that it is included in a dictionary, it is best to keep the two words separate in formal works. For some, whether the writing is formal or informal, the word anyday will always be seen as a typo. See the two examples below for the same publication using the term in the same purpose and spelling it two different ways.
When it comes to fighting fakes, Alibaba’s head of Internet security says co-operation beats the courtroom any day. [The Hindu]
Afternoons get busy, as terraces and homes fill up with the aroma of spices and oil, and women of the house get to work to serve those who prefer homemade pickle to branded ones from the supermarket anyday. [The Hindu]