It seems like it would be better1 to simply issue weekly refund checks of overcollected sales tax to people who qualify. Then the merchants have to track ID2 but they don't need to know your income or deal with a separate tax transaction.
1 For values of 'better' equal to 'if this horrible idea were considered desirable at all'. 2 But only on the transactions they want considered for the refund, perhaps. Or alternately, you could simply have the tax bureau accept authenticated receipts and issue refunds on that basis.
Another option, less overhead-intensive, would be to exempt the poorest from all sales tax, and have a card you show the register to establish examption. A food-stamp card you're paying with would automatically qualify. Of course, groceries are already exempt from sales tax in any halfway sane jurisdiction, so never mine.
Actually, everything is already exempt from sales tax in any halfway sane jurisdiction (there are just very few halfway sane jurisdictions). It seems to work fine for Oregon, so let's just eliminate it instead of trying to patch it. :)
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1 For values of 'better' equal to 'if this horrible idea were considered desirable at all'.
2 But only on the transactions they want considered for the refund, perhaps. Or alternately, you could simply have the tax bureau accept authenticated receipts and issue refunds on that basis.
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Pardon my typo in the previous comment; I'd fix it if I could.