Doug ... When we bought a PU truck for our son in New Hampshire - with no sales tax - and lived in MASS, the dealer charged the sales tax for MASS and gave us a check from the dealer made out the MASS RMV for the tax.
Lots of MASS residents try to buy 'stuff' in NH because of the no sales tax. Matter of fact, there is a huge mall right over the border above Boston and it gets lots of NE MASS sales for that very reason.
SO, NH vehicle dealers charge the tax and give the pre-paid tax check ... sales tax needs to be paid in MASS at the time you get an RMV plate, not at the dealer, although in MASS, dealers can often register the vehicle and have plates in stock - it saves the buyer that trip to the RMV.
If the dealer charges you any tax, I'd just make sure you get credit for that when you register it in Missouri.