It’s a monosyllable we so associate with complaining that it’s a verb we use for complaining, as well as a noun we use to describe a complainer or a complaint. Sometimes we even see it in print to represent the sound of discontent: grump, grump, grump.
The word grump isn’t as old as you’d think, given how apt it is to describe a person who is given to complaining. Our earliest written record of it is actually in the plural, and generally in the phrase humps and grumps to refer to slights or other indignities:
But though they [the elderly] are suffered to live, ‘tis under many hardships and restrictions, many humps and grumps, and scarce a day but they are asked what they do out of their graves.
— Daniel Defoe, The Protestant Monastery: or, a Complaint against the Brutality of the Present Age, 1727
From the written evidence we have for the phrase humps and grumps¸ our best guess is that it was a colloquialism that was common in speech long before it showed up in print. Grumps eventually dropped humps and gained its own life in the early 1800s to refer to a bad mood. (This meaning was the basis for the verb grump, which appeared in the 1800s as well). This grump also seemed to be informal:
Hannah had the grumps, for being up late didn't suit her.
— Louisa May Alcott, Little Women, 1868
And so it remained, even when grumps dropped its plural -s and was used to refer to a person in a foul mood. One of our earliest pieces of written evidence for this particular meaning of grump is from a dictionary of dialect terms from the turn of the 20th century.