If a mother of (removed) girls and (removed) boy becomes handicapped and dependent, who is required to take care of her? Her son or her daughters? They are all married with children. I was told the son is obligated to take her in first because the husbands of the daughters are not obligated to take care of their mother in law. But we could make the same argument for the wife of the son.

When a mother faces such a predicament in life, it becomes incumbent on all her children to help her and take care of her physically and emotionally as much as they can. Due to the son being the stronghold of the family after the father, it is his duty to care for his mother even if his wife is not able to. In Islam, the sons wife should regard her mother in law like her mother and take as much care of her, but if for some reasons she cannot or does not wish to do so, then it is upon her son to hire someone to take care of her if he himself cannot take up the heavy responsibility to take care of his handicapped mother. The reward is immense and so is the satisfaction not to mention the heartfelt prayers done by a helpless mother to her children.
As for the daughters, they too should share some kind of responsibility if they have support from their spouses and families and if it doesn't hinder her other responsibilities. The same reward would also go for them.
Be sure that to leave a mother facing such a helpless state and not receiving help from her children will surely earn the wrath of Allah.
The children should remind themselves when they were young and helpless that it was their mother who bore many difficulties to raise them. This is a test from Allah to see if they're performing their duties well enough now.
Lastly, they should also remember to be kind and benevolent to their mother who also needs emotional support apart from being helped physically. InshAllah it will be easy for them with the help of Allah.