Can Payeezy API manipulate FirstData Payeezy Global Gateway Hosted Order Page payments?

Yes the Payeezy Developer APIs can process a refund for a transaction that was processed through a hosted payment page. You would have to ensure that you capture the transaction tag and authorization number from the POST back in the hosted payment page response to perform the refund through the Payeezy Developer APIs. If you're getting invalid transaction tag it means that the transaction tag is not associated with a transaction that processed with that MID/Merchant token. For further review please provide the correlation ID so we can access logs.