I believe New York State requires collecting sales tax on shipping charges. When I setup sales tax, it appears that the tax collected is only on product prices and does not tax the shipping charge. Is there a way to set it up to also charge tax on shipping cost on GoDaddy Online Store?
I don't see how you can add a tax just for the shipping amount in GoCentral.
Your options would be to add the amount to your prices or add another tax: Settings > Taxes > Add another location. In both cases you would have to use averages -- which I know is not a solution.
In this case, GoCentral is probably not the platform for you as it is limiting in some areas. This is one of them.
Maybe consider looking at WordPress/WooCommerce which allows you to have this level of tiered taxation.