Her sister has Down’s syndrome; does she have to offer expiation because of not fasting?

Firstly: 

Down’s syndrome is a
genetic disorder; it includes delays in the development of cognitive ability
and physical growth. The external signs of Down’s syndrome include a crease
in the eyelids, short neck and hands, and poor muscle tone. 

Secondly: 

For fasting to be
obligatory, it is stipulated that one should be of sound mind, because of
the words of the Prophet (blessings and peace of Allah be upon him): “… The
Pen has been lifted from three: from the insane person whose mind is
deranged until he comes to his senses, from the sleeper until he awakens and
from the child until he reaches maturity…” Narrated by Abu Dawood, 4399;
classed as saheeh by al-Albaani in Saheeh Sunan Abi Dawood. 

If your sister is affected
by a delay in cognitive ability to the extent that she has not been able to
reach the level of discernment, and she does not understand the teachings of
Islam, then in this case she is not obliged to fast or to make up missed
fasts, and you do not have to pay the fidyah on her behalf, because she is
not accountable in the first place. 

Shaykh Ibn ‘Uthaymeen (may
Allah have mercy on him) said in ash-Sharh al-Mumti‘ (6/324): Anyone
who does not have cognitive ability is not accountable, and none of the
obligatory duties of Islam are binding upon him, such as prayer, fasting and
feeding the poor instead of fasting. In other words, he does not have to do
anything at all, with the exception of financial obligations. End quote. 

And Allah knows best.