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:
"How Amazon spins its sticky threads all through the cloud."