Is the wife sinful if she does not give her husband from her private money to go to his family, knowing his need for money, and that she bears part of the household expenses and has the right to dispose of her private money?

The husband is obligated to provide for his wife, even if she is wealthy. The wife is not required to spend from her own money on the household, unless it is done willingly, in which case she will be rewarded. Similarly, she is not required to give her husband from her money, and if she refrains from doing so, there is no sin upon her. It is her right to spend her money on what Allah Almighty has made lawful.