In Campaign mode you'll only be using 1 rider meaning your going to need to grind a ride with the 39 remaining riders in the game. All are unlocked from the beginning. Below is a list for you to mark off as you go along.
Brennon Eldred
Chase Outlaw
Claudio Montanha JR
Cody Campbell
Cody Nance
Cody Teel
Cooper Davis
Dakota Buttar
Dener Barbosa
Derek Kolbaba
Eduardo Aparecido
Emilio Resende
Fabiano Vieira
Gage Gay
Guilherme Marchi
J.B. Mauney
Jess Lockwood
Joao Ricardo Vieira
Jorge Valdiviezo
Jose Vitor Leme
Kaique Pacheco
Keyshawn Whitehorse
Kurt Shephard
Luciano De Castro
Luis Blanco
Marco Eguchi
Mason Lowe
Matt Triplett
Nathan Burtenshaw
Ramon De Lima
Robson Palermo
Rubens Barbosa
Ryan Dirteater
Silvano Alves
Stetson Lawrence
Stormy Wing
Tanner Byrne
Troy Wilkinson
Valdron De Oliveira
Winston Lopez
So go in to Arcade mode, pick a rider, a bull and then an arena. Be sure to survive for the full 8 seconds to ensure the ride is completed. Keep repeating being sure to change rider each time.
Remember there is no achievement tracking for this achievement so be sure to not miss a rider as you make your way across the list because you wont know who you missed out on. I was unable to find any stat either to show which riders you've rode with.
Lastly it is worth looking at my solution for the achievement involving riding on ever arena in the game in order to hit 2 birds with 1 stone.