There's plenty of evidence that can lead one to a conclusion on how the kids died. Tim looks physically fine, and could pass for alive, if not for the heavy, almost painful sounding breathing you hear from him throughout the game. It's possible that this relates to his death, and if the gas mask item is considered, it can be assumed he suffocated. This could be due to smoke from when the Weaver home burned down or something like an asthma attack. Tom's is more straightforward; his body is mangled as if he fell from a high place, possibly the bridge on the map. This is backed up by the fact that his leg brace is found under the bridge during 1AM. Plus, the audio clip from Lillian when you pick up the leg brace ends with "we all fall dead", further evidence that he fell and died from the injuries sustained. Lillian herself floats through the air in a way that looks like swimming, and two of her items relate to the well, one being the bucket and her stuffed bunny being found there. This leads to the idea that she fell into the well and drowned. Caleb is implied to have been buried alive, as shown by his behavior in-game, and one of his items is a broken shovel. It is also implied by a poem found in No Way Out that Mary is the one who killed him, as a line states "my mother killed her little son". Finally, Benny is implied to have died in his sleep, as he spends most of his time during the night sleeping, only attacking the player if they wake him.
WMG / Wick
The Weaver Kids' Deaths