The title of your questions seems to answer it!
What would I be tempted to do?
To purchase the vehicle in State B, of course, and then drive it home to State A.
How might State A discourage my plan?
By imposing a use tax:
https://www.investopedia.com/terms/u/use-tax.asp
(PS: This website is based in the UK. While we're happy to help people from around the world, there probably aren't any regular contributors here who're familiar with US Federal and State laws, so the amount of assistance we can offer you with such questions will be limited. The UK got rid of its sales tax, known as Purchase Tax, in 1973, replacing it with Value Added Tax. As we don't have different tax structures in different parts of the country, there can be no use taxes here).