Due to the very nature of grey literature, it can be one of the most challenging to search and find what you're looking for but it is literally everywhere! Luckily, a large portion of documents by government organizations can be easily found using Google Scholar. You can also take advantage of Google's source search which also works well for finding freely available data sets!
Google Scholar will bring back scholarly and grey literature in your searches such as government or university technical reports, bulletins, and sometimes newsletters. You can adjust the year of publication to bring forward only those that are current or within the timeframe you wish to explore.
Using Google.com to your advantage
Searching basic google but adding site: .gov will pull results from governmental websites, site:.org will pull from organizations*, and site:.edu will pull from institutions of education. For example, I might want a fact sheet on the endangered leatherback sea turtle in Georgia. I could search in Google "leatherback sea turtle georgia site:.gov" and expect to find listings from Fish and Wildlife, NOAA or even National Parks System.
*Organizations can have a wealth of grey literature as well but it is important to check for reliability like with anything.
Federal Agency Websites
These will often have individual state information for the topic you're looking for and can often times be searched within the website itself. Federal Agency Websites that may be helpful for your class:
U.S. Fish and Wildlife Services : https://www.fws.gov/
U.S. Geological Services : https://www.usgs.gov/
National Oceanic and Atmospheric Administration : https://www.noaa.gov/
National Parks System : http://www.nps.gov