Spock said (in episode Dagger of the mind) that because they have disposed of emotion there is no need of violence. Extending this then we could assume that there's no bullying among Vulcan children because there is no emotion. But, is this true? Are Vulcans born already with the ability of extremely controlling their emotion? Would something like this not need training? Which means that Vulcan children might be emotional and some could be bullies? It's a possibility. Especially if we consider that Romulans and Vulcans are genetically closely related, and that Romulans are emotional and many of them are violent. Wouldn't it be logical then to assume that Vulcan children are also emotional but then become less so with age, training and cultural/social rules?
Greg Cox is not the only one interpreting that Spock was bullied as a child by the other full blooded Vulcan kids. The 2009 movie writers also think the same (I know, I know, a lot of people think the Kelvin timeline movies don't carry weight

but....)