The responsibility for the events is primarily on Mr. White, but some can be attributed to his wife and son as well as to Sergeant Major Morris.
Although Morris knows that two people have had bad experiences with the paw and the wishes it granted, he passes it along. Instead of destroying it on his own, he brings it to White's house to destroy it there. For that reason the fault is partly his.
White proves unable to resist the idea of three wishes, even after hearing the major's creepy tale. He pulls it out of the fire and uses it despite the possible danger. He continues to use it after things start to go awry. Therefore, he is responsible for what happens.
Herbert suggests that his father wish for money, which he seems uncomfortable with, but does so anyway. That puts some weight on the son.
The mother begs White to wish for their son to come back to life but doesn't specify, as he was before the accident. The final horror can thus be partly laid on her.