Can a debt collector pull my credit report?

yeah they can. The collectors review CR to find the person that skipped payments. The collectors look for recent updates like new address, phone number, recent payments, recent transaction etc

Collection agencies can pull your credit report if you have outstanding debts. Checking your credit report for the purpose of debt collection is permissible under the Section 604 of the Fair Credit Reporting Act (FCRA). Your credit report is allowed to check by credit grantors, collection agencies, insurance companies, employers (but only with permission), mortgage lenders, landlords, utility companies, student loan lenders, government agencies, and judgment creditors.