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.