“Why did the chicken cross the road?”/“To get to the other side” is a classic riddle from the 19th century. A rubber chicken version is:
 
Q: Why did the rubber chicken cross the road?
A: She wanted to stretch her legs!
   
“Why did the rubber chicken cross the road? She wanted to stretch her legs!” was posted on the newsgroup alt.org.toastmasters on October 14, 1998.
 
“Why did the clown cross the road?”/“To find his rubber chicken” and “Why did the rubber chicken cross the road?”/“To get to the rubber side” are other “rubber chicken” riddles.
     
   
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. 
     
Google Groups: alt.org.toastmasters
Rubber-chicken circuit
TLRDESOTO
10/14/98
Consider the statement: “Some Toastmasters aspire to participate in the rubber-chicken circuit.“What’s the genesis of the term “rubber-chicken circuit”? Refers to the standard menu fare of rubbery chicken at eat and speak engagements?  Or, the use of rubber chickens as props?  Any good stories you want to share?
 
Oh, BTW.  Why did the rubber chicken cross the road?
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She wanted to stretch her legs!
 
Terry Robinson
Red Bird Toastmasters
Duncanville, Tx
   
1 April 1999, Houston (TX) Chronicle, “Take my joke, please” by Mike McDaniel, Entertainment sec., pg. 1:
MIKE McDANIEL, Houston Chronicle TV Editor. Houston Chronicle; Houston, Tex. [Houston, Tex]01 Apr 1999: 1.
Why did the rubber chicken cross the road? To stretch its legs.
 
Google Books
Jokelopedia:
The Biggest, Best, Silliest, Dumbest, Joke Book Ever
Edited by Ilana Weitzman, et al.
New York, NY: Workman Publishing
2000
Pg. 50:
Why did the rubber chicken cross the road?
She wanted to stretch her legs.
 
Twitter
James Henry
@olerafa
@adamgould
Why did the rubber chicken cross the road? ... She wanted to stretch her legs!
7:32 PM · Mar 21, 2009·Twitter Web Client
   
Google Books
The Mammoth Book of Really Silly Jokes:
Humour for the whole family
Edited by Geoff Tibballs
London, UK: Constable & Robinson Ltd
2011
Pg. ?:
Why did the rubber chicken cross the road?
She wanted to stretch her legs.
     
Facebook
Flames Charcoal Chicken is at Flames Charcoal Chicken.
March 21, 2017 ·
Why did the rubber chicken cross the road?
She wanted to stretch her legs.
 
Google Books
Hilariously Lame Jokes!:
800 of the Best Worst Jokes That Get a Side-splitting Laugh! (Or at Least an Eye-rolling Groan!)
By Harry Hilton
Berkeley, CA: Ulysses Press
2018
Pg. ?:
Why did the rubber chicken cross the road?
She wanted to stretch her legs.
 
Twitter
Clucksthechicken
@cluckschicken
#TuesdayThoughts #lifeneedslaughs
Why did the rubber chicken cross the road?
Coz she wanted to stretch her legs🐔🐔🐔
3:10 AM · Mar 26, 2019·Twitter for Android