Is it permissible for a woman to appoint someone else to stone the Jamaraat on her behalf, if the Jamaraat location is far away from her?

Praise
be to Allah.

Firstly: 

The basic principle is that if a person is able to stone the Jamaraat for
himself, it is not permissible for him to appoint someone else to stone them
on his behalf. 

Shaykh Ibn Baaz (may Allah have mercy on him) said:

if he is healthy, he does not have the right to appoint someone else to do
it on his behalf; rather it is obligatory for him to stone them himself,
because when he entered ihram for Hajj, it became binding upon him to
complete it, even if it was voluntary, because starting Hajj makes it
obligatory to complete it, as Allah, may He be glorified and exalted, says
(interpretation of the meaning): “And
perform properly (i.e. all the ceremonies according to the ways of Prophet
Muhammad (blessings and peace of Allah be upon him)), the Hajj and Umrah
(i.e. the pilgrimage to Makkah) for Allah”
[al-Baqarah 2:196]. 

The same applies in the case of ‘umrah, as is indicated in the same verse:
if a person begins it, it is obligatory for him to complete it and perfect
it, and he does not have the right to delegate some of the actions of Hajj
so long as he is able to do them, according to the correct view.

End quote from Silsilat al-Fataawa ash-Shar‘iyyah – Fataawa al-Hajj,
compiled by Khaalid al-Juraysi (p. 107) 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: 

It is not permissible for a woman or anyone else to delegate someone to
stone the Jamaraat  on her behalf, because stoning the Jamaraat is one of
the actions of Hajj, and Allah, may He be blessed and exalted, says
(interpretation of the meaning): “And
perform properly (i.e. all the ceremonies according to the ways of Prophet
Muhammad (blessings and peace of Allah be upon him)), the Hajj and Umrah
(i.e. the pilgrimage to Makkah) for Allah”
[al-Baqarah 2:196]. 

For this reason the Prophet (blessings and peace of Allah be upon him) gave
permission to the weak ones among his family to leave Muzdalifah at night so
that they could reach Mina before it became crowded, and then stone Jamrat
al-‘Aqabah, but he did not give them permission to appoint someone else to
stone it on their behalf. Similarly, the Prophet (blessings and peace of
Allah be upon him) gave permission to the camel herders to stone the
Jamaraat on one day and not the next, but he did not give them permission to
appoint someone else to stone them on their behalf. This confirms that
stoning the Jamaraat must be done by the pilgrim himself (or herself). 

End quote from Majmoo‘ Fataawa Ibn ‘Uthaymeen (23/107) 

Secondly: 

If a person is unable to stone the Jamaraat for a valid reason that is
acceptable according to sharee‘ah, such as illness or weakness that makes
him unable to bear the difficulty of stoning the Jamaraat and putting up
with the crowds, or because he is elderly and is not able to stone the
Jamaraat at all, or a pregnant woman fears for her foetus, and similar
excuses, then in such cases it is permissible to appoint someone else to
stone the Jamaraat on one’s behalf. 

Ibn Qudaamah (may Allah have mercy on him) said: If a man is sick or
detained or has a valid excuse, it is permissible for him to appoint someone
to stone the Jamaraat on his behalf.

End quote from al-Mughni (3/257) 

Shaykh Ibn Baaz (may Allah have mercy on him) said: There is nothing wrong
with appointing (someone to stone the Jamaraat) on behalf of one who is sick
or a woman who is unable to do it, such as one who is pregnant or heavyset
or weak, and she is not able to stone the Jamaraat.

 End quote from Majmoo‘ Fataawa Ibn Baaz (17/301) 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:

if the pilgrim cannot stone the Jamaraat, either because of old age, or
because he is sick or because she is a pregnant woman, or because he or she
is blind and will find it difficult, then in this case there is nothing
wrong with appointing someone to do it on one’s behalf, because it is a case
of necessity.

End quote from Majmoo‘ Fataawa Ibn ‘Uthaymeen (23/119) 

In order to evaluate cases of weakness or difficulty for one who is
accountable but is not able to bear it, reference should be made to the
situation of the individual and his honesty before his Lord. You know
yourself best, and can compare your case to the details mentioned above. If
you did that because you thought it most likely that you were too weak to
bear this difficulty, or because someone whose knowledge you trust advised
you to do that, then there is no blame on you for what you did, in sha
Allah. 

And Allah knows best.