I looked up public libraries in Jacksonville and in Green Cove. It does not look like you meet Jacksonville's requirements to get a library card there (see their website), but you could contact them and ask. Green Cove (Clay County Library) does not list requirements for card eligibility anywhere easily found on their website or newsletter; you can contact them by phone or e-mail and ask about their requirements.
My e-library (a public library in California) issues cards to any verified California resident, but physical presence is required once a year to renew the card. That card is the key to e-services.
Local taxes usually fund local libraries. It is not realistic to expect libraries to offer all services free to everyone. I am a retired librarian. Even my former employer does not now provide me with e-services. Part of our lease agreement with the e-library vendor is that e-services are offered only to currently enrolled students and currently employed staff. Public libraries have similar vendor agreements worded for their particular clientele.
