Of American-English origin, the noun grease monkey (plural grease monkeys) is an informal appellation for a mechanic, especially one who works on motor cars or aircraft.

The noun monkey has long been used—frequently with modifying word indicating the occupation concerned—to denote a person engaged in a trade or profession, especially a person performing a subordinate or menial task, or one which involves physical agility. For example:
– the noun powder monkey, attested in the mid-17^th century, designated a boy employed on a ship to carry gunpowder from the powder magazine to the guns;
– the noun road monkey, attested in the late 19^th century, designated a person employed to repair logging roads.

The earliest occurrences of the noun grease monkey that I have found are as follows, in chronological order:

1-: From a letter from San Pedro, California, that Claude Newburn, a sailor in the U.S. Navy, wrote to his parents on Sunday 22^nd September 1918—letter published in The Lindsay News (Lindsay, Oklahoma) of Friday 4^th October 1918:

I have my application in for change of rating frm [sic] seaman to gas engine mechanic, and am almost sure I will pass. If I pass will get shipped to some place in Virginia for three months school. That will not be bad for when Uncle Sam is through with me I will make some Garage a good “grease monkey.”

2-: From the following advertisement, published in The Idaho Republican (Blackfoot, Idaho) of Monday 18^th April 1921:

When Your Car Needs Repairing
To get the best and cheapest results. Look for these things.

[…]
THE MECHANIC—Is your job going to be done by a real mechanic, or a “grease monkey” under the casual supervision of a foreman. How much real experience has he actually had? And Where? Has he worked in the leading shops and in the large service stations, where they use best methods and short cuts that are unknown to the average small shop mechanic.

No Man Can Do Good Work Unless He Knows How
You will find the proper qualifications at the shop of
Sisung & Flaherty
WITH ANDREN AUTO COMPANY

3-: From an article about the U.S. racing driver Jimmy Murphy (1894-1924), published in The Tacoma Daily Ledger (Tacoma, Washington) of Sunday 2^nd July 1922:

Success to Jimmy has come after years of hard work. If you knew him when he first started the game of speed you will know that every bit of the rewards he is now reaping have been won by work. Today he is the same likable chap that he was in the days when he was a “grease monkey,” as driver’s mechanics are dubbed. The glory of the game has not turned his head in the least particle.

4-: From the following advertisement, published in The Ardmore Daily Press (Ardmore, Oklahoma) of Sunday 1^st April 1923:

Anderson’s Auto Supply
Phone 287
NIGHT OR DAY SERVICE
A complete line of Parts and Accessories that we can recommend to you
Our Gasoline Tests 58.6 Gravity
MOBIL AND VELVET OIL SERVICE
Repair Department in Charge of
PORTER CRAIG
(Cadillac Grease Monkey)
8-10 East Main St.

5-: From The Long Beach Press (Long Beach, California) of Sunday 10^th June 1923:

Brambila Completes His First Year in Business in Long Beach

Recording a notable and brilliant page in the history of the commercial life of Long Beach, the J. L. Brambila organization has just celebrated the first anniversary of its advent into the local motor car world.
The event was appropriately celebrated by Mr. Brambila and every member of his sales, service and clerical forces, from Sales Manager Edwin Elliott down to the newest “grease monkey.” Congratulations from all over the country poured in on Brambila, and leading officials at the Maxwell Motor car factory in Detroit sent him a huge floral offering to express their appreciation of his efforts.

6-: From The Houston Post (Houston, Texas) of Sunday 25^th November 1923:

WATKINS JOINS BUICK FAMILY AS A SALESMAN

Another veteran of the automobile business in and around “these diggin’s” joined the Brazos Valley Buick company recently.
His name is D. L. Watkins, but it seems that nobody knows “Barney” by his rightful and lawful cognomen.
Barney made his “debut” in Houston’s automobile business about 12 year ago as a grease monkey in a garage. He worked as a mechanic up until about three years ago, then decided on the white collar and tie.
Getting into the sales end of the game, Barney made good with another Houston firm, and now he hangs his hat over with the Buick company.