Beach Buggy Racing 2: Island Adventure (UP)

After you beat the final race in Adventure mode, you will receive a clue about a key. It is found in Glacier Gulch, the third event of the Firework Fury Cup (Ice-10). About 10-15 seconds into the lap, veer off the road to the right at the first opportunity. Drop down into the crack for the key (and a shortcut).

Now you can open the secret gate to the right of Castle-5. Press twice to access the Pumpkin Patch. Get all 27 stars for the 100% and the achievements will unlock.

I'm unsure if you can get this earlier or if the key only appears in that specific race, but this is how I did it.