The title isn't too clear so I'm just posting this to clarify what you actually need to do...
Invite friends / Randoms into a crew and then start a mission in crew coop. you can play the same race 10 times or different ones. I believe "The Initiation" is the shortest race but I preferred to play "The Getaway" as it was just as short and more fun.
Note: You can just hit square at the end of the race to start the race again, it will still count towards your count of 10.
The trophy will pop at the start of the tenth race.
Feel free to comment any suggestions