Here is the full list of all 12 VR Karts trophies - there are 8 bronze, 3 silver and 1 gold.
Win the Pro cup
Win the Amateur cup
Record a lap time in Time-Trial
Win a race in Turbo mode
Play with alternate controls
Win the Rookie cup
Win a race
Customize character
Complete tutorial
Compete in an online race
Win an online race
Win every race in Rookie, Amateur and Pro Cup championships