Airborne,
If Missouri state law allows you to own a suppressor, then you are good to go. You still have to purchase the suppressor through an authorized NFA/class III dealer.
To do that, you have to fill out the proper paper work, have it signed (sort of approved) by you chief law enforcement officer, and send in the paperwork to the BATFE with the required $200 fee. Once approved, you will receive a form 4 with the tax stamp affixed to it. That proves to the dealer that you have paid the fee and are now authorized to take possession of the gun.
That's the condensed version, but you get the idea.
Happy shooting.