I've just tried and this is what I got (I live in San Jose): "You must be a resident of Los Angeles to register for a library card."
How do I get a library card?
"To receive a library card you can fill out an application in person at any Los Angeles Public Library location, or download the application form , fill it out and bring it in to any location, or pre-register over the Internet by visiting the Library's Web Pre-Registration Page. If you only want access to online resources, and are a resident of the City of Los Angeles, you can get an eCard online. Library cards are free for California residents."
[1] https://www.lapl.org/faq/how-do-i-get-library-card
Looks like they allow you to get a library card in person at the library if you're not living in LA. They just don't allow you to get the card online. I got my Card when I visited their main library in LA.