Horses that vote “neigh” (nay) is an old joke. “If a horse were in Congress, how would he vote? Neigh!” was printed in the Alabama Beacon (Greensboro, AL) on August 3, 1850. “How do horses vote.—Neigh” was printed in the Bennington (VT) Banner on June 24, 1869.
 
“Why can’t horses run for Congress? They always vote neigh. #ClassicJokeFriday” was posted on Twitter by Ellen DeGeneres on June 5, 2015. “Why are horses lousy at voting? They only vote neigh” was posted on Reddit—Jokes on June 17, 2018.
 
           
3 August 1850, Alabama Beacon (Greensboro, AL), “Wit and Humour,” pg. 4, col. 1:
If a horse were in Congress, how would he vote? Neigh!
 
24 June 1869, Bennington (VT) Banner, “Odds and Ends,” pg. 1, col. 4:
How do horses vote.—Neigh.
     
Google Books
Parties with a Purpose:
A Handbook for Activity Directors
By Shirley L. Barrett
Springfield, IL: Charles C. Thomas
1980
Pg, 26:
A horse’s whinny; and negative vote. (neigh; nay).

Google Groups: alt.fan.pratchett
Rant w.r.t. Sect. 28
Miq
2/9/00
(...)
>Of course, having a horse in parliament would be difficult, as they
>would always vote ‘neigh’.
 
And this would be a bad thing because…?
 
Twitter 
Derek Demoranis
@ThisDerekHauk
How come horses can never agree on the rules of the race? They always vote neigh.
6:53 PM - 7 Jun 2009
   
Twitter
Andrew Emerson
@blandyblemerson
A horse legislature would never accomplish much. They’d vote “neigh” on everything
12:37 PM - 21 Jun 2011
   
Twitter
Greg Janda
@GregJanda
Why shouldn’t you ever elect a horse? Cause they always vote Neigh. #SundayPunday
9:11 PM - 16 Sep 2012
 
Twitter
Matthew Hartmann
@Matt_Writes
How did the horse vote? Neigh! #horsejoke
6:03 PM - 13 Apr 2014
 
Twitter
w e l l - l i k e d
@IsaacLeicht
Horses always vote neigh.
12:09 PM - 16 Apr 2014
   
Twitter
Robert Sephazon
@Sephazon
Why is it difficult for horses to ever get things approved?
Because they all vote neigh!
10:15 PM - 3 Apr 2015
 
Twitter
Ellen DeGeneres
@TheEllenShow
Why can’t horses run for Congress? They always vote neigh. #ClassicJokeFriday
7:57 PM - 5 Jun 2015
 
Twitter
Pun Manuel
@ThePunMANuel
Why can’t a parliament of horses accomplish anything?
-They always vote “neigh” on everything #puns
12:21 AM - 2 Dec 2017
 
Reddit—Jokes
Why are horses lousy at voting?
submitted June 17, 2018 by heratio90
They only vote neigh.