I know this is an old question, but since no one else has answered it...
I did a quick Google search for "libraries for non-residents" and found a nice Wikipedia link that lists a number of libraries that you can access even as a non-resident.
The free sites that are mentioned are:
- Digital Public Library of America
- The Library of Congress
- Open Library
Open Library is probably the closest thing to a real "library" of the three. It actually has books--rather than allowing you to read manuscripts, etc.
The Wikipedia site also has a list of USA libraries that offer cards to non-residents, with each charging an annual fee. The listed libraries are:
- Brooklyn Public Library
- Charlotte Mecklenburg County NC
- Enoch Pratt Free Library, MD
- Fairfax County, VA
- The Free Library of Philadelphia, PA
- New Orleans Public Library, LA
- Orange County Library System, FL
All of the libraries in the last list use the Overdrive app, which you will have to download, but is a free app. I do almost all of my reading using the Overdrive app nowadays, and I love it.
If you do decide to pay for a library membership, make sure to look at the information on the wikipedia site, as it shows how many Overdrive titles are available at each library. Please note that in my experience, certain genres are more likely than others to be part of a library's e-book catalog--frequently books I am interested in reading are not available or have a long waiting list. Make sure to check out the options for putting yourself on a waiting list, or for requesting new books. I have been really successful with both.
P.S. I am sorry that I didn't include more links, but as a newbie to this stack exchange site, I don't have enough reputation.