It would be up to the individual car dealer if they would sell you a car in San Diego without a full, valid California license. An out of state permit may be enough for some automobile dealerships but you would need to contact them directly to find out what exactly would be required of you to buy a car.
If you are able to purchase a car with your NY learner's permit, you can get a San Diego car insurance quote here with us. For information on how to register and title a car in California, contact the CA Department of Motor Vehicles.