The best place I found to do this was on the first map, "The Bog".
There is a circular area on the west side of the map, south of the Fuel Station that has a number of mushrooms there.
They look like this, make a squelching / popping noise when you run them over, then burst and disappear.
I don't know if there are enough in this area to earn the achievement in one run or if you'll have to quit and restart the map. When you run over enough of them the "Mushroomer" message will pop up on your screen and some music will play. Congrats!
Note - Sometimes the Xbox tracker takes a long time to update with this game, it may not update progress or pop the achievement until the next time the game autosaves. If you are certain you've run over enough of them or just want to check your progress, drive out of the area until "saving" appears on the bottom of the screen and it should update or unlock.