Yes, he can. But, there are several issues to understand.
1. Someone who can be claimed as a dependent by another taxpayer, can't have dependents of their own, even if the person who could claim them, won't claim them. In this case, if you and your spouse have no taxable income, and don't file a tax return, then you are not "taxpayers", and so your child is not disqualified from claiming his siblings, even though your oldest child probably would qualify as your dependent.
2. In order to claim his siblings as dependents, they must be younger (which seems to be true) and your oldest child must have more taxable income than you as the parents (this also seems to be true in your case).
3. It is not necessary that your oldest child actually provide more than half the support for his siblings, it is only necessary for this test that your younger children not provide more than half their own support. However, it is necessary that your older child live in the same home with the younger children for more than half the year. This is called a "qualifying child" dependent, even though the siblings are not his children.
4. The credit for claiming a "qualifying child" dependent is up to $2000 per child, but the credit can't be used to create a tax refund if the son owes no tax. There is a secondary version of this credit called "additional child tax credit" which is based on your child's income earned from working. He would have to have income from a job or self-employment to get anything for claiming a dependent, and he won't come close to getting the full $2,000 per child unless his taxable income from working is well over $20,000.
