Can I legally move out at 16 in the state of Missouri?

No. You have to be 18.

No, you can not move out at 16, in Missouri you have to be 18.

At 16 or 17 you would be arrested and normally put into a detention facility for children, until arrangements can be made to return you home.

Also at 16 you can't legally enter into a contract so you can't sign a lease. And anyone allowing you to live with them could be prosecuted for parental interference, harboring a minor etc.