Having grown up in a nearby metropolis (Tulsa, OK), and married to a person who grew up in one on the opposite side of that state (Cinncinnati, OH), I can tell you that outside of Missouri, almost everyone in neighboring states pronounces it with a long-E sound.
I had relatives in the extreme Southwestern county (Macdonald) of Missouri, and they pronounced it with the "uh" sound at the end instead. As a result, I learned to code-switch: Inside Missouri it is "Missouruh", and outside "Missouree". So I'm fairly certain that pronunciation is just something residents use locally to identify fellow residents (I've observed this phenomenon in more localities than I can mention. Most prominently New Orleans, which residents pronounce more like "Nawlins").
As far as dialect goes, Missouri is an interesting case. You could simplify and just say the state speaks American Midland, which is probably the closest live dialect to "Standard American English". However, the truth is not quite so boring. St. Louis has its own dialect, but otherwise the state exists on the border between the North Midland and South Midland dialect. My SW Missouri relatives definitely were in the South Midland camp.
University of Missouri English professor Donald Lance made an impressively complete study of the high vowel vs. schwa issue. It turns out that its use in the confines of Missouri is even more complex than outside (so thankfully you didn't ask about inside the state). But it does appear that the original introduction of that pronunciation is most likely a feature of the South Midland dialect area. As of the 1960's it was in fact still pronounced that way by some folk outside the state.
However, younger people (I'm guessing those growing up with access to national media via radio and TV), largely quit pronouncing it that way.
So if you find yourself a non-Missourian South Midland speaker born before 1930, there should be a good chance you will hear "Missourah".