They can do that, as local governments (other than cities) have no statutory claim to sales tax.
If local government poobahs are convinced the threat is real, they will begin pleading with state electeds to raise the tax and let the county keep the additional revenues.
Big County Czar Karen St. Hilaire Says Raise the Tax or the Towns Pay
The locals shouldn't fold so quickly. They should insist on a share of the new revenue in exchange for their carrying the water for higher taxes.
Of course the County could just keep the current tax rate and eliminate the share to local governments, forcing them to live within their means.
St. Lawrence is one of only a handful of counties still at 7%, and since it's run by Democrats, they understand the need for all residents to "pay their fair share."
St. Lawrence County leaders want sales tax hike | NCPR News