What is the ruling on a wife refusing to perform the obligatory Hajj due to her children not being taken along, and is she sinful for that?

Hajj is obligatory to perform immediately, and your wife must hasten to perform Hajj if she is able. Leaving the children with her family is not an excuse to delay Hajj, as long as they are safe, unless she has already performed the obligatory Hajj of Islam, in which case it is binding upon her.