Note this only works for 14 days. You can just make new accounts with new temp email so it's not that bad. Anyways
Go to https://www.nypl.org/library-card/new
and click on Get Started
When you are in there, it will ask you to enter your name,surname,email adress and dare of birth.
In Name and Surname just type fake name, you can find it either on internet or make it up.
Same with Birthday information.
Then in email you can just use Temp Email, one of best sites for it is https://10minutemail.com/.
After you do all of that click on next
Then it will tell u to give your New York Adress you can get fake one at https://www.fakepersongenerator.com/random-new-york-address-generator.
After you do all of that click next
Then it will ask you for alternative adress just leave blank click next
After that click next and then on next page click next again.
After that just type username you want and password you want and then click next.
After that you have your library card which lasts for 14 days after that just make account on https://www.overdrive.com/apps/libby and enter your library card number there.
And you are done.