How do I apply for a donated car?

Local Lions or similar.

You don't in most cases. Cars are donated... they are then auctioned off for cash that the charities then use for what they do.

I agree, I do not know any case, where a donated car actually goes to a person free. As smoothy said, a car to be actually donated (in the US) has to go to a company who has the license and registered to accept donations. In those cases, they are sold, and the money (minus the fees) are given to the charity. The companies who process these, make a profit, by the fees they charge. Often these are fairly high fees. In some areas, I know two places in Atlanta (when I lived there) that had car sales lots where they sold them like a used car lot instead of auction. I bought one there once.

Now with that said, you may check with the religious group of your choice and see if they may have any connections, also if a veteran check with several of the vet groups.

Most donated cars, 1. are very old, 2. very high milage, 3. often have several mechancial issues. At the auctions most do not even run.

I live in rural NW CT and there is a charity here that donates cars for free. I know a woman who wasn't even a hardship case who got one, because she was the only applicant who could/was willing to learn stick shift.
(I never would have known the program existed had she not told me. That's the way it is with many small charities.)

Most charities will require some type of financial records to prove need. However, even with a free car, you need tags and registration, insurance, money for gas, oil changes, repairs, tires etc. As Fr. Chuck noted, "Most donated cars, 1. are very old, 2. very high mileage, 3. often have several mechanical issues".