Jose Luis Hernandez rose from vegetable chopper to sous chef at an exclusive New York restaurant -- and saved $100,000 along the way. Recently, the illegal immigrant from Mexico contacted real-estate agents in Brooklyn, N.Y., where he currently rents an apartment. "I wanted to use my money as a down payment on a house," says Mr. Hernandez, 32 years old. In doing so, he sought to join thousands of undocumented workers who in recent years have purchased homes using an Individual Taxpayer Identification Number, or ITIN, instead of a Social Security number. The Internal Revenue Service doesn't give Social Security...