I've tried my hand at Googling this answer through various terms, and have come up dry. Is there such a specific word for this field? If not, what would could be coined as such?
The answer is rather simple, and quite surprising.
Which according to M-W means: the study of idiom.
I don't know of a specific word, but idioms fall within the range of...
Phraseology: the study of set or fixed expressions, such as idioms, phrasal verbs, and other types of multi-word lexical units.
EDIT: A follow-up in a few dictionaries, including the OED, failed to corroborate Wikipedia's information. In light of that and John Lawler's response, I'll add the far more commonly used sense of 'phraseology'. Here's the OED definition (1a):
Phraseology: The selection or arrangement of words and phrases in the expression of ideas; manner or style of expression; the particular language, terminology, or diction which characterizes a writer, work, subject, language, place, etc. Occas. also: an instance of this.