Q:

Is a debt collector allowed to make a hard inquiry on your credit report?

Under what circumstances is this permissible? What is the purpose of this anyways?

A:

According to the Fair Credit Reporting Act:

any consumer reporting agency may furnish a consumer report [...] to a person which it has reason to believe [...] intends to use the information in connection with a credit transaction involving the consumer on whom the information is to be furnished and
  involving the extension of credit to, or review or collection of an account of, the consumer

See p12 (section 604). The usual interpretation of this that I've heard is that a debt collection agency that owns or has been assigned a debt can make hard pulls on your credit report without your consent. This link seems to support that (and references the same part of the act, among others):

According to the Fair Credit Reporting Act, [...], any business can access your credit history without your permission provided the business has a valid "permissible purpose." The FCRA notes that one such permissible purpose is to review your credit information in connection with the collection of a debt. Thus, if you owe money to a debt collector, the debt collector has the legal right to pull and review your credit report.

If they haven't been assigned the debt or own it outright, I believe you have a legal right to dispute it. Consult a lawyer if this is actually a situation you face. 
Once use for this is if the debt collection agency has trouble locating you; since your credit report normally contains current and past addresses, this is one way to locate you. 

