Its really just a software problem. The vast majority of products made in the US which you describe all have unique identification numbers called UPCs, Universal Product Codes. With the exception of in house made items like custom baked goods almost every product has a universal code. That code can easily be linked to a database which identifies its characteristics by state and county etc.
Once you have the characteristics by state/county/etc. (how the product is described legally), you can then cross reference it to where it is sold to and calculate a tax.
Is it a lot of data? Absolutely, but it is doable and the person/team who does it is sure to have a fairly large business charging companies who want their products sold online and companies using their system.
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There will likely be multiple providers which should drive the price down and economic profits to zero. There is no reason why you would pick 1 tax calculator over another except price.
Most foods won't be an issue since they aren't sold as much online.
If you are a retailer of a new item with low volume, that is really where this is a problem. Once you make over a million a year in sales, you will likely have to hire an expensive consultant to figure out your tax implications.
[Additionally Added] I agree that it will be a challenging transition. But, in 10 years, this will be a non-issue.