According to the Office of Social Security (See Resources for link), a social security number is not only crucial for a job, but also for receiving social security benefits and other government-sponsored services. Taxation is the primary reason for the issuing of social security numbers. Self-employment is a means of making money without a social security number; also, there are several types of small businesses that employ without asking for social security information.
Contact self-employed businessman such as painters and contractors because most of them do not require you to have a social security number. Ask them if there is any extra work they need assistance with.
Create your own job opportunities. Pick jobs that you can perform for others, such as lawn mowing, painting someone's bedroom or babysitting. Create your own business cards so that people can know that you are serious. Make yourself as accessible as possible.
Use your skills and talents to land jobs and make money. Write down a list of all your skills. Think of ways you can monetize your skills. Charge for piano lessons if you play the piano, sell artwork if your a painter, etc.
Setup garage sales and yard sales from your home. Find old things that you no longer use, like clothing and electronics to sell at your garage sales.