The hypothesis that these are attributive nouns and not attributive adjectives
One relatively common viewpoint seems to be that racist and sexist, when used to modify a following noun, are actually "attributive nouns" (or possibly appositive nouns). I.e. "a racist tradition" means "a tradition associated with racists" in the same way that "a family tradition" means " a tradition associated with families/with a family." Somebody mentions this explanation in this Reddit thread: Misogynist vs misogynistic?
If true, this would indeed explain the lack of the adjective suffix -ic in this particular context. However, there are many other contexts where racist and sexist are unambiguously used as adjectives. We can't say "a tradition that is family" but I can say "a tradition that is racist." We can't say "the most family tradition I've ever heard of" but I can say "the most racist tradition I've ever heard of." Same goes for sexist.
We can use racist as a predicate without an article: "That's racist!" If "racist" were a noun in this sentence, it would have to be a non-count noun of some kind meaning something like "racially prejudiced stuff." But it doesn't seem to work that way.
So I don't see any other option than to accept that for people like me, racist and sexist are indeed truly adjectives (at least in some cases).
And once we've established that they can be adjectives, I don't know of any reason why they wouldn't be able to be used as attributive adjectives before nouns.
Other adjectives (maybe) ending in -ic
Here is a list of other -ist words that have been called adjectives: http://www.onelook.com/?w=%2Aist&posfilter=a
Hot Licks pointed out in a comment a few words like this that seem even older, such as Methodist, Baptist and nationalist. Searching Google Books, I found a few examples of "Baptist" being used as a predicate in an apparently adjectival fashion from the 1800s:
But it also is Baptist all the same. ("Gleanings from the German Field," by Rev. J.C. Grimmell, from The Baptist Home Mission Monthly, Volume 9)
The book is very sour, very sour indeed–it is Baptist! (On a Review of Mr. Noel's Work on Baptism," by Philalethes, in "Correspondence" of The Baptist Magazine for 1849, Vol. 41)
Interestingly, I was not able to find any examples of "very Baptist" from this century where "very" was used as an adverb. There are more recent examples, though:
It was a very American book and a very Baptist book, and it went
to the heart of Mullins's religious and theological concerns. ("Edgar
Young Mullins," by Fischer Humphreys, in Theologians of the Baptist
Tradition, edited by Timothy George and David S. Dockery)
Also, I did find an example from around 1888 of "very Methodist," where apparently "very" is an adverb and "Methodist" is an adjective:
They laid their arms across each other's shoulders in a very Methodist
way ("Au Large," George W. Cable, The Century, Volume 35)
So it seems that this overall phenomenon of "-ist" words being used as adjectives is more than a century old.
Many of these words actually do have explicit adjectival counterparts ending in -istic. Unfortunately, I don't have a good explanation for why others do not. The Oxford English Dictionary's entry on "-ist" merely states:
Many of the nouns in -ist give rise to adjectives in -istic suffix,
-istical suffix; but words of modern formation are to a great extent used adjectively unchanged, as in the royalist party, a Bonapartist
plot, nonconformist principles.
Possible origins of "racist" as an adjective
Influence from French
I was sorry to see a recent answer to this question was deleted, as I thought it had some valuable information. It mentioned that the earliest use of "racist" cross-linguistically was in French. Basically, it seems that the word racist as an adjective may come from the French word raciste. This word could already be used as a noun or an adjective in French. The reason seems to be that in general, French words ending in -iste don't have to receive a suffix to be used as adjectives. For example, the French adjective corresponding to English "moralistic" is moraliste, according to WordReference Dictionary.
The Oxford English Dictionary considers "racist" to have been used as an adjective in English since early on: its first citation for the noun use is in 1924, while the first for the adjective use is from 1927, only 3 years later. This supports the idea that racist may have come into English as both noun and adjective, rather than starting out as a noun and gradually coming to be used as an adjective.
However, the three earliest OED citations for "racist" as an adjective are unfortunately technically ambiguous, as they are examples where the word modifies a following noun, and so conceivably "racist" in these quotes could be seen as an attributive noun. The earliest OED citation that provides an unambiguous case of the word being used as an adjective is actually from 1979. Somebody might be able to find earlier examples on Google Books, though.
Reinterpretation of the noun in compound words
Even if racist actually was originally only used as a noun in English, it seems like it could easily come to be used as an adjective because of the aforementioned ambiguity between the "attributive noun + noun" structure and the "adjective + noun" structure.
Many of the words ending in -ic refer to followers of some school of thought or people who take some particular position. And it's natural for words like this to be used as attributive nouns. To me, it seems possible that the adjectival usage evolved from earlier attributive-noun uses. (Something similar seems to have happened to the word fun, which is an adjective for many people today.)
Some kind of semantic difference from other -ic words
It might be relevant that racist and sexist both group together in meaning as well as form; unlike most -ist words, their main meaning currently is "prejudiced based on [race or sex]." Another similar but less common word is classist: it would be interesting to see how many people use it as an adjective as well.