Visit their home pages. Have they digitized materials? If they haven't digitized, have they published Finding Aids (which give you a good sense of what materials are contained in their collections)?
If you're striking out on finding primary sources, then look for secondary sources. Journal articles and books that discuss your topic will hopefully have bibliographies directing you to their primary sources (and then you can evaluate whether you can obtain/read those sources).
When searching a database of books, whether the catalog or Google Books or WorldCat:
Newspapers and other periodicals from the Revolutionary time period can also be primary sources for your research.
The Google Books digitization project makes pre-1923 books, which are in the public domain, available for download: