Yes, I would definitely recommend amending your taxes if you are not able to obtain the EIN or SSN for the person who provided care for your disabled brother. It seems that the information must have flowed from your federal to your state return if they are requesting a copy of your federal return. It could be possible that the dependent care credit didn't apply to your federal return if your tax liability was already zero, but did apply to your state return. If that is the case, you would only need to send in the amendment for your state return.
Please see these TurboTax Help Articles:
How do I Amend my return for this year?
How do I Amend my return for a prior year?
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