Can't help you on Missouri, but I've noticed these questions pop up with some frequency.
It would be a real resource if somehow we could put together a state-by-state guide based on rodders recent experiences.
Anyhow, in Ohio, there is info on the DMV and EPA web sites.
The DMV uses the State Highway Patrol to do an inspection, that is mostly confirming that the parts aren't stolen. Roadworthiness is largely determined by the builder/owner.
The EPA, I believe, goes by the model year of the engine (if it can be proven) to determine the pertinent regulations and if emissions testing is required.
Another concern would be insurance. In Ohio, the title says something like "homebuilt" or "self assembled". That covers a very broad range - from building a stock equivalent out of parts from various cars to something completely from scratch. Other than liability, any other coverages may be difficult to obtain. I haven't yet researched that aspect.
I haven't found anyone in the process who actually confirms that the vehicle meets any roadworthiness criteria. You would think that would be where the DMV would get involved, but they apparently don't.
One aspect that is kind of weird, is that (in Ohio) you bring the completed vehicle for inspection. Of course, you can get in touch with inspectors beforehand, but there's no sort of plan approval, as there would be for building a home.