The short answer:
Yes, for specific family members that meet Zakat conditions, and who the Zakat giver is not already obliged to provide for. So scholars unanimously concur that a man cannot pay Zakat to his parents, or to his wife, or dependent (young) children; though a wife can (some scholars say “should,” in terms of higher divine reward) give to her husband. Zakat may appropriately be paid to all other close relatives that qualify for it, according to the most endorsed and best supported juristic opinions.
The detailed answer:
There are three test questions to determine if you can pay Zakat to a Zakat-eligible family member:
Who is dispensing your Zakat payment to the family member, you or an authority/agency?
Are you as a Zakat giver already obligated in Islam to provide for the living expenses of the family member?
Is ownership of the wealth out of which you are paying your Zakat continuous with the family member receiving the Zakat payment?
Answering each of these questions will tell you if it is permissible for you to pay your Zakat to a Zakat-eligible family member.
Who Is dispensing the Zakat payment to the family member, you or an agency?
Let us remember that a fully functioning Zakat system has an authority assessing, collecting, and dispensing Zakat payments. In such a case, any relative of yours that the Zakat authority determines Zakat eligible may receive Zakat payments from it, even if you have paid your Zakat into it and regardless of your financial responsibility for that relative, including your wife, parents, and dependent children.
Many of us now pay our Zakat to nonprofit organizations, our own Zakat Foundation of America, for example. In such a case, if a family member receives Zakat payment from the agency we pay our Zakat to, any family member that is Zakat-eligible may receive Zakat from the foundation, even though you have paid your Zakat into it.