Most companies will not approve you as a shopper without a SS#. They are obligated to use due diligence that they only work with folks legally allowed to work in the country. And should you earn more than $600 in a calendar year they are required to send you a 1099. While you are obligated to pay taxes on any and all income earned as a shopper, most companies will not let you do even a couple of jobs without having your SS#. It also helps them avoid authorizing as a shopper folks they previously terminated for whatever reason who want to sign up again under an alias.
Alternately, some companies want an EIN, but in my area a EIN has too many other issues with getting it to be worth it for the few jobs I am likely to perform for those companies. And since we are not allowed to subcontract our work to others, there is no need for an Employer Identification Number.
Generally if you found the sign up site from a legitimate source (not an email link), the chances are that it is a legitimate company with at least rudimentary safeguards for your sensitive information.