I know that weben means to weave or to spin. But no such word as durchwebt. Source: Wie Amazon die Cloud durchwebt
This is a play on words that won't work in English.
On the one hand, the author is referring to the web as in world wide web.
On the other, he/she is using the German word "durchweben", which refers to weaving, specifically adding special (usually gold) thread to produce decorative, precious cloth. In English, this would be "shot through (with gold)". I suppose the author is ignoring the "precious" aspect here, and instead focuses on the fact that these threads are spread all through the fabric - with the implication that this is not entirely welcome only becoming clear in the context of privacy protection.
Edit: Also, he/she may well be playing with the association of a spider sitting in a corner and sneakily spinning its threads to catch unsuspecting prey - a rather obvious connection in this context :)
So as far as I can see, what the author is aiming at is something like:
I'm surprised not to find the direct translation for German durchweben in any of the answers thus far, so here you go. The English word is:
That is the direct translation of the German word. It exists both in German and in English, so why not give the translation to explain the meaning.
A possible synonyms (or at least word close in meaning) would be: durchflechten
The particular meaning in the phrase "Wie Amazon die Cloud durchwebt" would be closer to "How (much) Amazon penetrates/pervades/controls the cloud", conveying that it was unknown up until now how much influence Amazon has in the area of cloud services. So from that perspective I am with falkb and his/her answer that the meaning is closest to "pervade" (also upvoted for that very reason).
The term "cloud" referring to something that existed for quite a while under a number of acronyms (such as SOAP) but is a prime example of brandwashing and Marketingese.
Oh and one more note concerning the connection to "the web". I'm sorry, but I can't see that. It cannot be ruled out, but I don't see anything in the text that substantiates there is a wordplay intended here with "the web" (as in the user-facing part of the Internet), commonly shortened to "das Netz" in German (and a literal translation of "the web").
Wordplays mostly play on the pronunciation, especially across languages, and this simply wouldn't work with "the web". If you would write the pronunciation of "web" for a German native speaker would end up as "Wepp", where the "e" is a short vowel and the "pp" is a plosive. Whereas the German verb "weben" is a long "e" and a proper "b" (not a plosive).
So I would hold it's not a wordplay, even though I cannot rule it out with 100% certainty.
"Durchweben" is a compound of "durch-through" and "weben-weave". It means to extend a web, so that it becomes ubiquitous without being perceivable.
I would translate it rather freely as
(maybe a bit flowery)
It's meant in the sense of
Yet another answer. To me the German one has a naturalistic tone to some extend, a tone that contradicts the cold technological world of the internet.
I think durchweben is not necessarily negative. I haven't read the whole article but I think translations like "dominate" may add something that isn't there. However, if there is indeed this focus in the article a title could be:
and so on.