I checked a number of reference works that focus on slang and didn't find anything in them related to the use of grunge or grungy before the 1960s.
Both the Fifth Edition (1961) and the Eighth Edition (1984) of Partridge, A Dictionary of Slang and Unconventional English, have no entry for grunge or grungy; but Beale, Partridge's Concise Dictionary of Slang and Unconventional English (1989) has this:
grungy. Dirty, unwashed and smelly; squalid: orig. pop music world, early 1970s, > more gen. (Hibbert, 1983.) Paul, 1982, 'She walked into the clubhouse feeling grungy and uncomfortable [having just woken up on a golfcourse, without knowing how she got there]'. Perhaps a blend of grotty and gungey; cf. scungey.
Neither Wentworth, American Dialect Dictionary (1944), nor the First Edition (1960) of Wentworth & Flexner, Dictionary of American Slang, provide coverage of either term. But the Third Edition (1995) of Chapman & Kipfer, Dictionary of American Slang, has entries for both words. Here is the one for grungy:
grungy (GRUHN jee) adj 1960s teenagers Shabby; squalid; dirty; =GROTTY, SCUZZY: I put down in my grungy little notebook that Max Frisch was a wise man—C Vetter/ ... the peerless, fearless, slightly grungy Grodin to investigate—Richard Grenier [origin unknown; perhaps sound symbolism resembling gross, mangy, mung, stingy, etc]
The Fourth Edition (2007) of Kipfer & Chapman essentially repeats its predecessor’s treatment.
Lighter, Random House Historical Dictionary of American Slang (1994), likewise has entries for both grunge and grungy. The entry for grungy includes two definitions:
grungy adj. [prob. fr. SCRUNGY] 1. Dirty; shabby; grimy; disreputable; unpleasant. [sixteen citations, starting with two from 1966, omitted] 2. Inferior; poor; dismal [citations, from 1969 and 1985, omitted]
And Ayto & Simpson, The Oxford Dictionary of Modern Slang (1992) has this for grungy:
grungy adjective mainly US Grimy dirty; hence, of poor quality, unappealing; unpleasant, bad; untidy. 1965—. Dirt Bike I would like to know who made those blasted white pants so popular—mine are splattered with oil specks and other grungy stains (1985). [Apparently arbitrary formation, after grubby, dingy, etc.; cf. GUNGE noun,]
The earliest instance of grunge and grungy that I’ve been able to find in the senses suggested by the foregoing definitions is in “Selected Daffynitions from Our Unabashed Fictionary” in Geochemical News, issue 32 (June 1962), near the end of the newsletter:
Grunge: A geologist who has been in the field for at least one week without a bath, without changing clothes, and without shaving. When a person has attained this state of affairs, he feels grungy.
That instance antedates the earliest OED citations (noted in FumbleFingers’s answer) by about three years.
Also supposedly before 1965, from the Congressional Record (1964, date not confirmed) [combined snippets]:
No more research funds are available until the ban on explorations is lifted, but the Conns are eager to continue in the cave as long as they can because of the bone-deep excitement of the work they are doing.
Jan, in speaking of their work, says Jewel is more than “a pair of grungy spelunkers" can ever explore completely, and is hopeful of enlisting full-scale expeditions to carry the work forward.
What a large assault by a corps of professional geologists and spelunkers will force Jewel to reveal is a matter for speculation. What has been found so far makes Jewel one of the most exciting and beautiful caves in America, and if any cave can outdo superlatives, Jewel seems the most likely candidate.
The occurrence of grungy in two geological settings prior to 1965 is at least an interesting coincidence, and may indicate (relatively) early usage of the term independent of the U.S. teenagers who adopted it during the 1960s—though the situation is complicated by the occurrence of the term grunge-jumper (twice) in Jack Kerouac, Dharma Bums (1958).
But I haven’t been able to find any reference to grungy in the sense of “envious” or “jealous” aside from the echo-chamber instances of recent, citation-free glossaries online. At this point, it seems to me, the burden of proof is on the purveyors of those glossaries to produce evidence that any such understanding of the term existed during the 1920s.
UPDATE (7/20/14): Source of the Online Glossary That Includes "Grungy: Envious"
I've found what may be the original online glossary that identifies grungy as a 1920s slang term meaning "envious": The Internet Guide to Jazz Age Slang. Though this glossary unfortunately doesn't identify any examples of 1920s use of grungy, it does (unlike every other version of the glossary that I've found online) specify the sources that it drew its entries from:
Note: the majority of the entries were gleaned from a great slang dictionary called Flappers 2 Rappers, written by Tom Dalzell (Merriam-Webster, 1996). This is the resource for those interested in slang from any decade of the 20th century. The reader will find more Jazz Age slang, along with literally hundreds of other words and selected etymologies. Details can be found at the Merriam-Webster site here [link omitted].
Many entries have also been added from The Writer's Guide to Everyday Life from Prohibition through World War II, by Marc McCutcheon. This book is an indispensable guide to all those minutiae of life during one of the most story rich periods in history. A must have for those interested in the Twenties! Check it out (along with all the other books in the Writer's Guide series) here [link omitted].
I don't have copies of either of these books, but they are the obvious next places to check for information that may authenticate the "grungy = envious" equation.
UPDATE (7/28/14): Published Reference Work Citing 1920s Usage of 'Grungy'
I obtained a copy of Tom Dalzell, Flappers 2 Rappers: American Youth Slang (1996) and found the following entry in the chapter on "The 1920s: The Flapper":
That's it—no citation of any published instance of the term, no specific (single-year) date for the first occurrence of the term as used in this sense, no specific prior authority listed as the source of the information. At the end of the section on 1920s slang, Dalzell does identify three primary sources for the slang terms included in that chapter:
The primary sources for this chapter were "A Flapper's Dictionary," in The Flapper, Volume 1, No. 2 (July 1922); The Flapper's Dictionary: As Compiled by One of Them (Plattsburgh, New York: The Imperial Press, 1922); and "Flapper Filology—The New Language," in the Philadelphia Evening Bulletin of March 8, 1927.
The first of these primary sources is available online courtesy of the bookflaps blog, but that list has no entry for grungy, though it does have one for grummy ("In the dumps, shades or blue"). Likewise, "Flapper Filology"—which evidently was published in 1922, not 1927, since Dialect Notes (1922), volume 5, number 5, reproduces its contents—doesn't include an entry for grungy. I'm less sure about whether I've correctly identified the third of Dalzell's primary sources, but "The Flapper's Dictionary As Recorded by Ella Hartung in 1922" may be that source. As the single-sentence introduction to that list observes:
This isn't the official [Flapper magazine] Flapper's Dictionary, but a version recorded in 1922 by a young woman who would - much later - become my grandmother.
The Hartung glossary doesn't list grungy either.
So we have definite matches for two of Dalzell's three primary sources and a plausible match for the third—and none of them list grungy as 1922 flapper slang. It is certainly possible that the term grungy arose later in the 1920s; but if that's the case, Dalzell doesn't give us much of a clue about where he came across the term.
Though Merriam-Webster published both Flappers 2 Rappers and another, similarly citation-thin volume by Tom Dalzell titled The Slang of Sin (1998), its association with Dalzell hasn't led it to expand on its own relatively narrow treatment of grungy:
grungy adjective 1 : shabby or dirty in character or condition 2 : characteristic of grunge music or fashion ... origin unknown First Known Use: 1965