Quite simply, the "problem" in How to Eat Fried Worms is the challenge for Billy to eat 15 worms in 15 days. In this way, the conflict can be seen as Billy vs. friends (and sometimes even Billy vs. parents).
Early in the plot, Billy and his friends (Alan and Tom) agree on a bet. Billy instigates the bet by saying that he can pretty much eat a couple bites of anything, even if he doesn't like it. It is Tom who actually suggests fifteen worms in fifteen days for fifty dollars. The first issue arises with how big the worms have to be in order to be eaten.
The bigger and juicier, the better!
The boys agree that the worms must be supplied by the opponents (and not by Billy). Billy uses an extraordinary array of condiments to help him stomach the worms, such as ketchup and horseradish. Unfortunately for Billy, his tummy gets quite upset. The continual ups and downs of Billy's stomach create suspense for the reader as we wonder whether Billy will be able to win the bet. Luckily, his parents and his doctor help him through the struggle and he eventually wins the bet!