Grape Allergy challenge requires you to finish the round with the least amount of grapes as possible.
This challenge might sound like an easy one but actually is quite hard: the game is built the way that you should always have some kind of grape income, be it bottom grapes, sour grapes or a top-10 bounty (check grapes page for more details). Also the only way for spending grapes is buying autopromote.
The ideal strategy would be to watch two ladders (your current and the previous) closely enough so you can snipe your multi or bias with someone promoting up/multiing. This will protect you from acquiring the bottom position for too long. Also, make sure to buy the most expensive autopromote on each ladder.
Winning is very possible since this challenge requires you to be a frontrunner. Although having little to no grapes each ladder makes your vinegar income very low and makes you weak to vinegar throws.
Yes, autopromote cost scales on each ladder, not necessarily the current top one, but the main reason is to keep yourself around the most populated area of the round.