Do the libraries have fax machines for the public to use?
The libraries do not have fax machines or scanners for public use.
Office supply stores and photocopying and print shops often have these services.
There are sites online that offer free faxing. However, they may put advertisements on your cover sheet or have page limits. The library at the University of Illinois - Urbana/Champaign describes features of 6 online fax services, and This blogger reviews 5 online fax services.