Jim's looks to be the most complete version I've seen. Once this whole virus is over with, anyone living near Virginia Tech, U. Georgia, Penn State, or Berry College (GA) can see if he's missing anything by checking at the library

The closing stanza appeared in dozens of college exchanges around the turn of the century. I know it's more than most people will care to read, but I think it's kind of neat to watch how the song changed over time. A few variations I haven't seen on here yet, that I've organized in order of year of appearance:

Yackety Yack, 1901, p. 333

The water dripped from the potato's eye,
The cabbage hung his head;
There was a sad old time in the kitchen that night
For the vinegar's mother was dead

The Picket, Feb. 1902, p. 11

The potatoes' eyes were full of tears,
The cabbage bowed its head,
There was grief in the kitchen that day
For the vinegar's mother was dead. -- Ex.

Bohemian, 1908, p. 148

The potato's eyes were full of tears,
The cabbage hid her head;
There was grief in the kitchen on that day,
For the vinegar's mother was dead.

Bethanian, 1916, p. 151

The water dripped from the potatoe's eye,
The cabbage hung down its head;
There was a sad old time in the kitchen that night
For the vinegar's mother was dead!

Captain Billy's Whiz Bang, Feb. 1922, p. 36

Cellar Ancestry

The potatoes eyes were full of tears,
And the cabbage hung its head,
For there was grief in the cellar that nite,
For the vinegar's mother was dead.

Tampa Bay Times, July 23rd, 1922, p. 4

Dim with tears
Were the old potatoe's eyes.
And grief made
The cabbage bow its head.
From parsnips and turnips came subbing sighs,
The vinegar's mother was dead.

The Chowanka, 1923, p. 82

The water dripped
From the dishes' eyes;
The cabbage hung his head;
There was a sad old time
In the kitchen that night
For the vinegar's mother was dead.

The Hutchinson News, Feb. 18th, 1924, p. 7

The potatoes eyes were red;
The cabbage hung its head;
There was grief in the cellar;
The vinegar's mother was dead.

The Elms, 1925, p. 116

A TRAGEDY

There were tears in the eyes of the potato,
And the cabbage hung its head,
There was a terrible grief in the kitchen
For the vinegar's mother was dead.

Marysville Journal-Tribune, Oct. 4 1928, p. 4

The potatoe's eyes were filled with tears,
The cabbage bowed its head.
The artichoke said never a word;
For the vinegar's mother was dead.

Retro, 1929, p. 99

There were tears in the eyes of the potatoes.
The cabbage hung its head
because the ears of corn had heard
that the vinegar's mother was dead.
[line breaks added by me for readability]

Casa Grande Dispatch, Jul 25 1929, p. 1

The potato's eyes were watering,
The cabbage hung its head,
For everyone in the cellar knew,
That the vinegar's mother was dead.

So we wonder if the beets, even though pickled, brought cauliflower.

The longest one aside from Jim's (and two odd ones I'll add in a separate post/s) I found is this:

The Wheaton Journal, Jan 6 1938, p. 4

The pumpkins were crying
And the old beets face was red
The turnip tops were dying
The onions' tears were shed
The potatoes eyes were full of tears
The cabbage hung its head
There was grief in the cellar that night
The vinegar's mother was dead.

You can see as time progresses, some vegetables come and go, the crying is attributed to other kitchen objects, and there's a slight change in wordage. The Harvard Lampoon of October 1901 Jim cited in 2011 was the earliest reference I could find, and even it borrowed it from someone else (it being in the "clippings" section").