Greylitsearcher is a tool that allows you to perform structured searches of websites using Google's sitesearch functionality.
These guides from other libraries also provide useful information on grey literature and tips on where to find it:
Grey Literature Guides – a directory of Grey Literature guides and other resources maintained by GreyNet
University of Groningen Libguide: Grey Literature
University of South Australia Libguide: Grey Literature and Other Sources
University of Toronto Libguide: Grey Literature
Applying systematic review search methods to the grey literature: a case study examining guidelines for school-based breakfast programs in Canada - by Godin et al. (2015): provides a helpful example of how one research team reported their multi-faceted grey literature search in a structured, transparent manner.
Searching for grey literature for systematic reviews: challenges and benefits - by Mahood et al. (2013): details how the research team carried out their grey literature search for a systematic review, and also discusses the challenges and benefits of including grey literature in a review.
Exploring issues in the conduct of website searching and other online sources for systematic reviews: how can we be systematic? – by Stansfield et al. (2016): provides and in-depth discussion of how to search websites systematically.