Does the American Mathematical Monthly allow ArXiv posting?
Yes, they do allow arxiv postings at any stage.  However, starting in 2015 the Monthly has moved to a double-blind reviewing system (which, for general readers of this site, is all but unheard of in the field of mathematics).  Posting on the arxiv before or during submission would of course cause a serious risk of ruining the anonymity of the author.

I happen to know through direct experience that the Monthly views such ruination as being the author's prerogative. 

It should be emphasized that this is a recent development, and whether it will be continued in the future is up to the board of directors of the MAA (and not the editorial board of the Monthly, a fact which I found surprising).

Added: Some remarks about the laissez faire philosophy of the Monthly's double-blind policy can be found here.  The difficulty of preserving authorial anonymity in an arxiv culture is mentioned.