Generally, you do this by retaining a private investigator, or by retaining a debt collection agency (which will add your debt to the debtor's credit report).
Post-judgment debtor's interrogatories posted by a judgment creditor to a judgment debtor can require the judgment debtor to reveal his SSN or the equivalent (e.g. an EIN or TIN).
It isn't clear why you would want their credit report, however. You have established that they owe you money and aren't deciding whether to extend credit which is what a credit report is used to do. A credit report does not identify the assets of a debtor, only their debts. This can be of a little help in the case of secured debt with assets of the debtor as collateral, but there are other public records that can be used to search the secured debt of a debtor (mostly real property records and Uniform Commercial Code records).