If you just make the product exempt, then there's no vouchers, no people to hand them out, etc
But if you make food exempt, then guess what? Mrs. Frisbee's Filet Mignon gets exempted too. There's simply no "fair" way to do it. If you make clothes exempt, A&F gets the same exemption as Rustler jeans. If you make toiletries exempt, Paul Mitchell shampoo gets the same exemption as Suave. Exempting a class of goods allows loopholes that can and will be exploited.
You can check income levels through social workers in private agencies without a "real" cost to the government. If people don't want to stand in line to get the exemption vouchers, well, then, they just don't get the exemption. Honestly, I wouldn't consider it worth waiting for.
I honestly believe that every wage earner in this country should pay some tax, even if it's just a buck. And honestly, thanks to hidden taxes, the vast majority of Americans DO pay some taxes. If we were to implement a national sales tax, I would rather skip the idea of exempting anything altogether than exempt a "class" of goods.