Searching the NYT and LexisNexis for "grade grubbing" and "grade grubbers" shows that there are two distinct usages from two different periods. It begins sometime in the 1970s and refers to college students whose sole focus is maximizing their GPA -- they forego having a social life, extra-curriculars, intellectual curiosity, and so on, in order to memorize more facts and study harder before tests. There's no suggestion, though, that this involves regular and predictable confrontations with their professors to complain about grades, and then haggle over what a truly fair grade would be.
"Grade grubbers" in the sense of pests who annoy teachers doesn't show up until the early 2000s. Presumably, the shift in the meaning was gradual, and so likely took place during the '90s.
Looking for the earliest account of an ongoing epidemic of student complaining and entitlement, I found this webpage / article from 1997. Its citations include contemporary news reports, academic articles, and first-hand stories from professors around the country. Part of it is concerned with the declining quality of the preparation that students arrive at college with, and the amount of work they put in when they get there. But it also features a lengthy review of student attitudes and behavior regarding fair grades -- by the time of the article, they'd begun to feel like just showing up to class and completing the assignments (no matter the quality) should secure a high grade, they demanded grade inflation in general, and in particular cases they would confront professors to have their test or paper score bumped up.
How far back did this go? Going through his list of over 100 citations, there are few from the 1980s, and none point to student vs. professor confrontations over grades. The only references to the decade are to the late '80s, and have more to do with falling achievement levels and growing apathy among students. Even the citations from 1990 to '92 don't support the view of "students as haggling pests." Not until the citations from 1993 to '96 do we see an explosion of that picture. Here's a quote that the author chose from a 1993 article in Harvard Magazine ("Desperately seeking summa"):
A student in a history course at Harvard “told … [the instructor] that he needed a better grade … because he needed to get into medical school. ‘I may not be very good at history,’ he said, ‘but I’m going to be a very good doctor, and I really reject the idea that you have the right to keep me out of medical school’.… Students are more likely to contest grades now. For many instructors, grading is the most distasteful thing they do. Often students help to make it unpleasant."
The author points out that the "student empowerment" movement of the 1960s can have little to do with the phenomenon of aggressive complainers since it far post-dates the heyday of the counter-culture. That's true of just about everything wrong with social relations these days -- they're a product of the 1990s, and were absent as recently as the '80s (e.g., political correctness, diversity sensitivity, multiculturalism, date rape hysteria, feminazis, homo enablers, etc.). Conservatives must wrap their brains around this fact of the timing, or else they'll cluelessly blame The Sixties and propose solutions that, whatever their worth in other respects, will do nothing to turn the tide against Millennial-era problems.
What about the link to widening inequality, and the attendant growing competition among aspiring elite members? We'd expect that kind of dog-eat-dog world to breed the aggressive grade-haggling that we see today. (See this article by Peter Turchin for a good review of the dynamics of inequality and its consequences.)
Income inequality began to grow sometime in the mid-to-late 1970s, and the higher ed bubble got going around 1980 (shown by percent of high school grads in college, and college tuition -- higher prices reflecting higher demand). Why the lag of 12-13 years from the start of the higher ed bubble until constant professor complaining about constant student complaining? I doubt it'd take that long to manifest itself, or for observers to notice. Compare the case of yuppies -- punks were already spray-painting "Die Yuppie Scum" back in the '80s, and an acclaimed movie from 1987 skewered the prevalent view on Wall Street that "Greed is good."
For that matter, forget the higher ed bubble. Once inequality starts to widen by the late '70s, why don't we read widespread reports of high schoolers -- or their parents -- pestering their teachers about boosting grades? The "grade grubber" in the '70s sense is still there -- Brian from The Breakfast Club, the nerd in Heathers whose contribution to talking about the suicide of a classmate is if they'll be tested on it later. Or that other grade-grubber in Heathers, whose sole purpose is maximizing his chances to gain early acceptance into an Ivy League school ("and please let it be Harvard," he begs God while praying at a classmate's funeral). But absent are any journalistic reports or fictional portrayals of high school kids confronting their teachers to haggle over grades.
One of the early-2000s references to confrontational grade grubbling in the press came from a high school teacher. My sister-in-law spontaneously remarked that complaints about grade-obsessed parents these days could have been said about her high school back in 1995.
So, it seems that hostile self-promotion is a product of both the rising-inequality trend and the falling-crime / cocooning trend. It can't be just the first, or else we would've seen it during the later '70s perhaps, but definitely throughout all of the '80s. And it can't be just the second, since I don't have that impression of the last stretch of cocooning from the mid-30s through the '50s. Students from that era are also bored, apathetic, dorky, and docile. But not hostile and self-promoting on top of it, like today's dorky student body.
That predicts that we'd see something similar going on among students of the Victorian era in Europe, and the Gilded Age in America. I have no idea whether that's borne out or not, though. They didn't have the same widespread level of public education and higher ed that we do during this incarnation of the "worst of both worlds" zeitgeist. But, it's something that someone could look into.