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There is a staturory framework in effect controlling the reporting of DOFD to the CRAs, and the CFPB has authority for administration of the FCRA, so yes, if the situation warrants, the CFPB can obtain the DOFD.
See FCRA 623(a)(5).
However, they cannot just contact a creditor and compel them to provide a DOFD.
First, they must have been under an FCRA requirement to have provided the DOFD to the FCRA, and not have complied.
An OC is only required to provide the DOFD on their account if they have first reported a charge-off or collection referral to the CRA.
They are then required to report the DOFD to the CRA within 90 days thereafter (section 623(a)(5)).
Similarly, if a debt collector has reported a collector, they are then required to obtain the DOFD and report it to the CRA within 90 days.
If the OC has previously reported a DOFD on the same debt, the debt collector can report that same date, but if the OC has never reported their DOFD, then the debt collector must contact them and attempt to obtain the DOFD.
If the debt collector has attempted to obtain the DOFD from the OC, but was unsuccessful, then the debt collector can report their best estimate of the DOFD.
There is no legal requirement that an OC must provide the DOFD to a debt collector, so the CFPB would be hard-pressed to enforce such a requirment.
That is why the statute provided for situations where the OC does not provide the DOFD.