Who are these locals? The Appalachian mounts run from Canada to Alabama. I suspect there are very few words of that length are pronounced exactly the same over that range.
LAY. My mother spent summers with relatives in the Smoky Mountains, which they thought were a range of the “a-puh-LAY-shuhn” mountains. Or something vaguely like that.
ap-uh-LATCH-un [ˌæpəˈlæt͡ʃɨn]
I was born and raised in Knoxville, TN. My parents are from Kingsport, TN. Knoxville is just outside the Smoky Mountains; Kingsport is in the middle of the ridge-and-valley Appalachians.
ap a LAICH e un Five syllables.
Wisconsin native speaker.
That’s probably the epi tome of wrong.
Appa-lay-chun. Cleveland.
App a LAY shun. My parents are Hoosiers, but I spent most of my childhood in Maryland, and went to college at Frostburg State University in Western Maryland. I never heard App a Latch un while in Frostburg.
And what vowel for the third syllable? An /ae/ (“latch”) or /eI/ (“lay”)?
There are at least three main ways of pronouncing it:
app-uh-LAY-shun (homophonous in my dialect with “appelation.”)
/ˌæ.pəˈleɪ.ʃən/
app-uh-LAY-chun
/ˌæ.pəˈleɪ.tʃən/
app-uh-LATCH-un
/ˌæ.pəˈlæ.tʃən/
(And there are apparently those who give a full two syllables to the “-ian” ending instead of schwa-ing it.)
(And the syllable division really should be more like ae-puh-LAY-shun, ae-puh-LAY-chun, ae-puh-LAE-chun, but since not everyone recognizes what “ae” stands for, I combined it with the following consonant[s] for the phonetic transcription.)
The first way is how I say it.
Exactly like I’d pronounce the word appellation (so -eɪʃən , -ay-shin)
As a foreigner, it ordinarily wouldn’t come up a lot for me, except the Appalachian Basin is quite frequently raised in geology courses.
I’m from Pennsylvania, near the famous trail. In these parts, folks say app-uh-LAY-shun. I never heard the other pronunciation until I heard it from Walter Cronkite, in a CBS documentary about poverty in app-uh-LATCH-ee-uh.
App-ah-lay-shun.
Grew up in Philly.
This is how I pronounce it as well.
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I’m in Maryland and around here I’ve always heard and spoken it “app-uh-LAY-shun.”
I was born in Baltimore, MD and moved to the Appalachian part of Ohio twenty years ago. All I heard in Baltimore was the first (long A) way, and all I hear in southern Ohio is the second (short A) way. I have adopted the local way.
Most of the time Lay. When talking about the college it’s Latch. I have found myself using both or either.
So out of curiosity, what would the actual English rule-based pronunciation for the word be? I mean, there has to be some kind of rule for whether that second ‘a’ is long or short, right?
I used to pronounce it more “A-puh-LAY-shia”, until I started playing Fallout 76. Now I pronounce it like the native I’m not 
Rules? There are no rules in English. 
a-puh-LAY-shuhn is closest (though my final vowel is a sort of u/i hybrid). New England born and bred.