The
MobileRead Wiki has a list of libraries that you don't have to appear in person or be a resident to get a card that gives you access to downloadable ebooks.
My experience is that Brooklyn Public Library is really on the ball with this...you sign up online, get your card number in an e-mail, and pay the fee online. You can be checking out books in less than 2 days from when you first sign up.
Orange County, OTOH, seems to have no connection between the computers that authorize library cards and their website. So, you sign up, and in 7-10 business days, you'll get a library card in the mail. Then, you use that to log into their site, pay the fee online, then wait until a human being gets the notice that you have paid the fee and flags your account as OK for borrowing ebooks. It took 20 days from the day I filled out the online form (which, BTW, requires you to fill in the Florida Driver's License number field, even if you don't live in FL...just put "none") until I could check out books online.
If you sign up for these two, you cover a lot of the books available through Overdrive.