Below is what I believe to be true about Missouri tax - any confirmation or rebuttal would be appreciated.
I have researched the issue on the following Missouri website: http://dor.mo.gov/motorv/salescalc.php As I understand it, if you are a Missouri resident and purchase a car outside of Missouri you DO owe state tax, however, you DO NOT owe local tax (city, municipality, etc.) UNLESS your local municipality has established a use tax. You can go the following website and plug in your info: https://sa.dor.mo.gov/mv/stc/default.aspx
You should not be paying Illinois tax on a vehicle if you live in Missouri. The reason Missouri dealerships are mad is that you can cross the river into Illinois and not pay local tax, which is typically around 2% of the purchase price.