Hello there,
Good job on figuring out how to assign a tax code to a customer! At this time, QBO doesn't have the capability to assign a tax code to a supplier. A common workaround is assigning the tax code to the product or service that's associated with the supplier. To do this click Sales from your left navigation menu > Products and Services > Edit the product/service associated with that customer. I've attached a screenshot with more details of where you'll see the option to adjust the taxes. Let me know if you have questions.