For Kant, morality only applies to rational beings. At some points, he will use the word "humanity" as a synonym. Thus, if we look at the second group of formulations of the categorical imperative we find:
Act in such a way that you treat humanity, whether in your own person or in the person of any other, never merely as a means to an end, but always at the same time as an end. (Groundwork of the Metaphysics of Morals)
For Kant, the only known rational beings are humans. God is posited as a rational beings but not known. Angels occupy a weird position as "pure limited rational being" which is absolutely never fleshed out.
The background behind this is that Kant believes the material world and animals are largely deterministic in terms of their behavior. They are limited to their natures. In contrast, human nature involves having rational and a free will that can choose based on reason. As a consequence, our nature is precisely to not be wholly determined by our biology (a point shared by Aristotle).
So, yes, spiders are not for Kant moral creatures and thus not bound to the categorical imperative. They can do no wrong (this isn't to say that they can't be involved in wrong -- like using spiders to torture people, but that they themselves are not moral agents there).