It's not illegial for the landlord to have the tenants social security number.
You can run a credit check on a tenant. But, you must have signed permission from the tenant to do so. The LPA has a good permission form to use for this. Also, be sure to get a copy of the tenants drivers license (make sure you make the copy and NOT the tenant).
The LPA also has a site for credit checks: http://www.thelpa.com/lpa/QuickCheck.html