Children have birthright citizenship in their parents' country/ies of citizenship. So the kids have birthright citizenship in their mother's country of citizenship. She can obtain passports for them from her country. Not all countries require both parents' permission for the child to obtain a passport or to travel. So she may be able to get them passports from her country, take them to her country.
US citizens are required to enter/exit US only on US passports. And US-citizen child must have BOTH parents' permission to get a US passport, and to travel. However, parents have been known to take their children back to their own country without permission of the US-parent. IF she takes them to her country without your permission, you could then file kidnapping, international abduction charges against your wife. She will end up in prison eventually & lose her green card, get deported to her own country permanently.
You fail to mention your wife's citizenship. So no way to tell whether her country will go after her if she takes her kids back to her own country. Her actions might be legal there. So yes, it's possible she can take the kids & leave you. Also possible you can't do much about it.
She's blackmailing you. Clearly, this marriage is over. You probably can only file for divorce and negotiate custody of the children at this point. Getting a court order of child custody, visitation & child support as part of the divorce settlement is all you can do. Unless you can prove she is an unfit mother, expect her to get custody. If you cannot work things out & agree to stay together, it's probably best she returns home to her own country, and try to divide custody if the children are old enough to travel, or else you visit them there.