I've always heard Missourah was the correct pronunciation and everyone always mispronounces it.
That might be a holdover from an old English pronunciation, similar to pronouncing the final a like a y as in Virginny (Virginia), Loretty (Loretta), Alabammy, et al. We used to hear this when we visited Appalachia, but the entire, once very distinctive Appalachian dialect seems to be disappearing rapidly.
My Missouri peeps (south-central) pronounce it Missourah. They also regard it as Midwestern.