"Writers' Contest" or "Writers Contest" [duplicate]

There are some interesting points here. One is whether the apostrophe (’) serves any purpose, but we can leave that for another day.

We speak of children’s books, not because, in this context, the books belong to the children, but because the books are for children. Children’s acts as a modifier rather than a determiner. As the ‘Longman Grammar of Spoken and Written English’ says,

[such] genitives have the role of classifying the reference of the head noun: the question answered here is ‘What kind of X? . . . In many cases, a classifying genitive is equivalent to an adjective or a noun modifier . . .

To take up that last point, children’s books could, just about, be replaced with juvenile books or junior books. We can certainly speak of adult literature as well as adults' literature.

I don’t think pets books would normally be found. Instead you might possibly see a section in a bookshop for pet books. A better example might be animal books (not animals' books), which clearly refers to books about animals rather than books for animals.

Grammar is as it is, not as you would like it to be.

A story book, a picture book, a fiction book are all grammatical, but a children book is not, at least in the sense you mean it (It is possibly grammatical in the sense of a book about children, but is not in common use even in that sense).