The Language of Salinger's The Catcher in the Rye (1984) says the muckle-mouthed means "talkative, garrulous".
"Muckle mouthed" did not originate from Salinger.
There was a historical person "Muckle-Mouthed Meg".
From Abbotsford and Newstead Abbey, by Washington Irving, 1835:
unfortunately, the young lady was hideously ugly, with a mouth from ear to ear, so that not a suitor was to be had for her either for love or money, and she was known throughout the border country by the name of Muckle-mouthed Meg.
See also the 1829 Tales of a Grandfather: Being Stories Taken from Scottish History:
You must therefore do a wiser thing, and, instead of hanging him, we will cause him to marry our youngest daughter, Meg with the meikle mouth, without any tocher
According to a genealogy she was:
Agnes "Muckle mou’d Meg" Murray of Elibank,
wife of Sir William Scott of Harden.
She was born in 1601.
On the other hand, a 5 September 1875 article titled Muckle-Mouthed Meg argues that the story isn't true.
However there is google document titled The Marriage Contract (14 July 1611) of Agnes Murray ("Muckle-mou'd Meg") (daughter of Sir Gideon Murray of Elibank, Treasurer-Depute of Scotland, 1612-21) and William Scott (son of "Auld Wat" Scott O' Harden, the Famous Scottish Border Reiver)
The only library copy seems to be at University of Glasgow.
Aside from Muckle Mouthed Meg, there is a line in The Last Rhyme of True Thomas (1894) by Rudyard Kipling :
I make honor wi' muckle mouth
(This poem refers to the real person Thomas the Rhymer who live long before Muckle Mouthed Meg)
Then, in 1944, shortly before Catcher in the Rye was written, in A spy for Mr. Crook:
After a tense moment the Member for Malvoisin let fly.
"Of all the flannel-footed, hump-backed, ginger-headed, muckle-mouthed, addle-pated sons of orang-outangs," he exploded. "How exactly like the authorities to pick on the wrong man."
So by the time Salinger used the term, it may have deteriorated into a general negative term, rather than the original meaning. In fact, a 1780 dictionary says that "Muckle" was already "Obsolete" and meant "Much".