The 1940s term “rubber chicken circuit” described the poor food at banquets, but when an actual rubber chicken was made and sold by novelty stores is not exactly known. Loftus International (begun as Loftus Novelty in Salt Lake City, UT, in 1939) is the leading seller of rubber chickens, but Loftus began selling the product in the early 1960s.
 
It is not clear that the rubber chicken novelty product existed and was popularly sold before the 1960s.
 
 
Wikipedia: Rubber chicken
A rubber chicken is a prop used in comedy. The phrase is also used as a description for food served at speeches, conventions, and other large meetings, and as a metaphor for speechmaking.
 
Comedy
A rubber chicken is a replica of a plucked fowl made from a latex injection mold. A popular sight gag and slapstick comedy prop, rubber chickens are sometimes used by comics as a mock weapon. They are also sometimes used by jugglers in place of clubs. The origin of the rubber chicken is obscure, but is possibly based on the use of pig bladders, which were inflated, attached to a stick and used as props or mock-weapons by jesters in the days before the development of plastic and latex.
 
One account attributes the first use of a prop chicken to John Holmberg, the Swedish black-faced clown of the early 1900s.
 
6 March 1935, Greensboro (NC) Record, “New York Day by Day” by O. O. McIntyre, pg. 4, col. 5:
Dining around Washington Square became a sort of rubber chicken hoax with an exploding cigar climax.
     
Google Books
On the Road for Uncle Sam
By Joey Adams
New York, NY: Published by B. Geis Associates; distributed by Random House
1963
Pg. 21:
This was followed by my big trick. Accompanied by much fanfareand much abra-cadabra-ing, I dove into the depths of my straw hat and produced a tired, wizened-looking, busted-up old rubber chicken.
   
Google Books
12 April 1971, Time magazine, “Life Among the Manson Jurors”:
Their boffo running gag: a rubber chicken purchased as a complement to Sheely’s chicken jokes.  The chicken made regular appearances in beds and toilets around the jurors’ hotel rooms at the Ambassador (cleverly dubbed the “Dambassador”).
 
Google Books
Esquire
Volume 75
1971
Pg. 48:
But in St. Louis last season, a verbose fan brought a blonde date and a rubber chicken to a Bruin game.
   
Google Books
Newsweek
Volume 87
1976
Pg. 150:
(“Is this campaign a turkey?’ he quipped when reporters jokingly presented him last week with a rubber chicken bought in a novelty store).
     
Modern Farmer
Farm Pop: Why the Rubber Chicken?
By Andy Wright on December 4, 2013
(...)
If anyone has had time to ponder the history and humor of the rubber chicken, it’s Jim Rose, president of Loftus International, a family-run Salt Lake City novelty company founded in 1939. Loftus International sells things like plastic dog doo, fart whistles and disappearing ink wholesale. But its most famous item is its rubber chicken.
 
Rose says they’ve been making rubber chickens since the early 1960s and it’s “unimaginable” how many they’ve sold over the years. The company sells between 10 and 20 thousand annually.
 
Loftus International doesn’t own a patent on the rubber chicken, and Rose doesn’t have any secret insider information on its origin.