Does she have to ask her husband who is travelling for permission to travel with her family?

The basic principle concerning a woman is that it is not permissible for her to leave her husband’s house without his permission. That is part of obedience to her husband that her Lord has enjoined upon her, and part of her care and respect towards him, and part of treating one another honourably. 

That applies except in cases of necessity; she may go out if need be. 

It says in Matalib Ooli’n-Nuha (5/271): It is haram for the wife to go out without her husband’s permission or unnecessarily, such as going to buy food if there is no one who can bring it to her. End quote. 

Please see also the answer to question no. 106150. 

As the husband is travelling and the wife (your sister) has her two children with her, her being with you is better and safer for her. If you travel and she goes with you, that is better for her than staying on her own with her children in her husband’s house with no one to keep her company. So there is nothing wrong with her travelling with you without her husband’s permission, unless he has told her not to travel, in which case she should ask him for permission and not go out without his permission, except in case of necessity. 

Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: 

If a woman knows that her husband will allow her to go to her family and relatives, is it permissible for her to go without his permission in case of necessity? 

He replied: 

This depends on what she knows of her husband’s preference. In some cases, the wife knows that the husband would give her permission to go out to see her relatives in case of necessity, and in other cases the wife knows that the husband does not want his wife to overstep the mark with regard to what he has permitted. So it depends on how the husband is. But if he has told her not to go out for any need or reason other than this specific purpose, then it is not permissible for her to go out except for this specific purpose. End quote from Fatawa Noor ‘ala ad-Darb by Ibn ‘Uthaymeen, 10/298 

He was also asked: 

What is the ruling on a woman who goes out without her husband’s permission? 

He replied: 

If her husband is present it is not permissible for her to go out except with his permission. If he is absent, she may go out so long as he has not forbidden her to go out and said “Do not go out.” If he has told her not to go out, he has the right to do that. End quote. 

Fatawa Noor ‘ala ad-Darb by Ibn ‘Uthaymeen, 10/298 

And Allah knows best.