There is plenty of precedent in general for reinterpreting problems as graph problems in order to obtain a different perspective. Correlation matrices of securities, for instance, can be interpreted as the weighted adjacency matrix of a graph. A common next step is to construct a minimum spanning tree from that graph, which can be used to build qualitative understanding of the market structure. See, for instance, here. Graphs offer several unique tools for analyzing data, with notions like betweenness that don't really have analogues elsewhere.
In your case, the graph would necessarily be bipartite. I don't know much about bipartite graphs specifically, but I do know that standard social network techniques aren't really designed for them. You might instead get better mileage out of website-website or feature-feature similarity matrices analogous to user-user and item-item similarity matrices used in collaborative filtering.
That said, I did find one interesting article that specifically uses bipartite graphs to cluster bivariate categorical data: "Bipartite graph partitioning and data clustering" by He, Ding, and Gu, 2001. (Free on Arxiv) I also found a paper that uses explicit bipartite graph structures for co-clustering: "Co-clustering by bipartite spectral graph partitioning for out-of-tutor prediction" by Trivedi, Pardos, Sárközy, and Heffernan, 2012. (Free) This opens up the possibility that you can apply a general bi-clustering algorithm to your data, which, like collaborative filtering, is well-established and might be more natural in your case.
As for statistical concerns specific to graph analysis, the easiest mistake to make is to forget that graph data is mostly not iid. It is more than hand-waving to conduct standard hypothesis tests on graphs; it is usually outright wrong, and will give you the wrong results. Instead, you need to use more sophisticated and computationally intensive techniques like the quadratic assignment procedure conduct hypothesis tests on graphs.