The idea of "thinking spiderwebs" is suggested by the title of an article: The Thoughts of a Spiderweb. Based on an article in Animal Cognition, they suggest that some spider webs are "at least an adjustable part of its sensory apparatus, and sensory apparatus, and at most an extension of the spider’s cognitive system." This, it is further suggest: "would make the web a model example of extended cognition, an idea first proposed by the philosophers Andy Clark and David Chalmers in 1998 to apply to human thought. In accounts of extended cognition, processes like checking a grocery list or rearranging Scrabble tiles in a tray are close enough to memory-retrieval or problem-solving tasks that happen entirely inside the brain that proponents argue they are actually part of a single, larger, 'extended' mind." Apart from the fact that the idea of an "extended mind" or "extended memory" does not originate with Clark and Chalmers, it is also questionable whether talk of "thoughts" is appropriate with regard to spiders, let alone with regard to their webs.
The author of the article knows these criticisms and suggests that we "more traditional theorists label these structures and spiderwebs alike as extended phenotypes, a term proposed by Richard Dawkins. Extended phenotypes are information from an animal’s genes that they express in the world. For example, bird nests are objects that are somehow encoded in the avian genome. And as with niche construction, natural selection affects the structure — different kinds of birds have evolved to build different kinds of nests, after all. But in the extended phenotype perspective, that selection ultimately just works inward, to tweak the controlling information in the animal’s genome."
I would put in my lot with the more traditional theorists, but I tend to agree with the critics who say "the only really strong case is the one with the most metaphysical baggage: us. “It is conceivable for cognition to be a property of a system with integrated nonbiological components.” I also agree that seems to be "where Homo sapiens is headed” or rather has been heading in this direction for a long time.
Luhmann's idea of "communicating" with Zettelkästen that has pre-occuupied this blog since its beginning is certainly one iteration of this idea.