I have the C&R app on my dresser to mail in.
Missouri changed the suppressor ownership law in Aug of 2008. You can now possess them if you have an FFL license. The C&R license is, so that suffices. Go the BATF website and request the application on line or by phone. Their hard copy is the only one they accept at the present. Fill out both sets, one for the BATF, one to mail to the Authority in charge of your specific legal jurisdiction. That's the responding agency if you dial 911, not the post office town name on your address. That jurisdiction gets the other copy of the BATF form.
You pay $30 for a three year license. Understand it allows you to buy C&R legal firearms and ship them directly to your home. You are required to log them into a "bound" book to verify they are on the premises, subject to the rare inspection by the BATF. It does not make you a FFL for a living, collecting sales tax, and dealing in non-C&R firearms.
License in hand, you can then purchase an suppressor, which requires going through the NFA hurdles of the BATF. That is a completely different procedure, where either the "Chief Law Enforcerer's" signature, or a trust set up and registered for the county are needed.
There are good reasons for a CLEO to refuse to sign - unfunded mandate, or simply never having to know what's out there if confronted by the media. That's not a bad position, as the BATF would be "to blame" for some future situation. Don't take it personally, especially if they don't even know you. It's been recommended and done to have a trust drawn up in Quicken, but many will insist it be a lawyer do it. You decide, suppressors aren't cheap once the price, $200 tax stamp, and the wait are all factored in.
Missouri does not presently have a regulation against using a suppressor when hunting. After all the hoop jumping, having an agent confiscate a suppressor, firearm, vehicle, plus any seat time in jail, it would seem to be irresponsible to also be ignorant of hunting regulations.
Step #1, C&R. Step #2, NFA application. Save your pennies up.