At what point did polyhedral dice become part of D&D?

David Wesely is credited by Dave Arneson with having introduced polyhedral dice to D&D. Theory from the Closet interviewed Wesely on a train to Gen Con in 2010 if you're interested in hearing about it from the man himself. (There are some sound quality issues with the recording, but it's quite worth it.)

This is from Wikipedia's article on Dice:

Seven- and eight-sided dice are stated in the 13th century Libro de los juegos to have been invented by Alfonso X in order to speed up play in chess variants.

Around the end of 1960s, non-cubical dice became popular among players of wargames,and since have been employed extensively in RPGs and TCGs. Reciprocally symmetric numerals like 6 and 9 are distinguished with a dot or by being underlined.

Gygax was was very familiar with these war games so he would have had access to these different kinds of polyhedral dice.

There is a much older game known as Little Wars by H.G Wells but I haven't found any evidence to support that it used dice

Also there is a much, much older game known as Kriegsspiel that also used dice.

For a strongly-sourced historical look at this question, see the analysis of the history of polyhedral gaming dice here. This story is basically about how the wargaming community became aware of the availability of polyhedral dice, about which companies adopted it first, and then how these dice ended being a part of D&D.

In short, though d20s were available in the 1960s through international standards associations, they were little known to gamers. That changed when Wargamer's Newsletter began publishing notices about them around 1969, describing d20s as a good means of resolving percentile statistics in games. Gary Gygax subscribed at the time, and mentioned that these dice would go well with a game he was working on called Tractics, a modern-setting wargame that made extensive use of statistics. In turns out that the cheapest way for Guidon Games (publisher of Tractics) to get the dice was to order a whole set of Platonic solids from a company in California called Creative Publications (one of their period advertisements is included in the blog post). From there, in June 1973, Gygax started posting about using all five Platonic polyhedra in games, in particular mentioning his work on fantasty campaign rules at the time.

All of this is excerpted from the chapter of Playing at the World on dice (Section 3.2.1.2) though the blog post has better pictures. The comments on the blog post address some of the alternative accounts (like Wesely's).

Incidentally, the Dalluhn Manuscript ("Beyond This Point Be Dragons") is remarkable for its lack of use of polyhedral dice - it is one of factors that makes it look like a pre-publication document rather than something later.

The original edition published used nothing but d6's as required dice, but had an option for use of d20's as well.

The 1975 release of Supplement 1: Greyhawk moved to the much wider use of polyhedrals, requiring d4, d6, d8, d10, d12, and d100 rolls.

There is limited evidence, however, that the earlier drafts may have extensively used percentile dice. Namely, a manuscript entitled Beyond This Point Be Dragons, supposedly found in the possessions of Professor M.A.R. Barker, and not quite authenticated by Dave Arneson.

There is little convincing evidence of polyhedral use other than the d20 and d100 prior to 1974 (the pre-release era), tho Gygax is said to have been using the d20 driven "Alternative" Combat system from before release. Dave Arneson did use percentiles almost exclusively in early drafts.

It is worth noting that the d100 rolls used 2d20, as the available d20's were marked 0-9 twice, not 1-20. The pentagonal trapezodipyramid was not commercially readily available until the later half of the 1970's.