Growing up in suburban Kansas City, we had both city, county and state sales taxes which could vary based on a short drive. And then there's certain entities which are allowed tax exemptions, like teachers making qualified purchases for some things.
Yes they can. Just have a constant price in that advertising region for the premium items you're advertising, eat the cost in locations with high sales tax rates (or rent, or wages, or shipping costs, or any other of the ten thousand other factors that they already deal with without issue), and make up for it in the areas where costs are lower than average.
Almost every single large business in the United States already does this, and it would be incredibly trivial for them to do it for sales tax as well.
Almost every large grocery store business does this, and a brief google street view tour of North St Louis will show multiple places where grocery stores have vanished and been replaced, if at all, by dollar stores.
the few times i remember handing over a specific sum of cash at thrift/vintage stores, indie shops, etc smaller stores here or there, they would always seem to include the tax in the displayed price if there was no computer to calculate the final tax, and the receipt (if any) was whole numbers throughout
So what's stopping them putting tax-inclusive prices on shelves or on stickers on the items themselves?