Don't want to put SS# on job application?

It has always been a firm principle never to share my SS# with any company until I have been hired, and it does no good sharing that information with anyone elsewise. Now, I'm strapped and looking for a job (Unfortunately min. wage), but so many corporations handle the entire hiring process online. This wouldn't be a problem, but many list SS#s as a required preliminary field before allowing someone to fill out an application, whereas on paper, someone can forgo such. Should I bother applying and abandoning my principle? Is it even federally legal for a company to require an SS# before an 'applicant' can even go through the application process?