Praise be to Allah
It is not permissible for a woman to donate eggs to be fertilised with the sperm of anyone other than her husband, even if the embryo will be implanted after that in the uterus of the sister of the donor. The prohibition on this kind of fertilisation was issued in a statement by the Islamic Fiqh Council in its statement no. 16 on Test-Tube Babies. In that statement it says:
In the session of the International Islamic Fiqh Council that was held during the third conference in Amman, the capital of the Hashemite Kingdom of Jordan, 8-13 Safar 1407 AH/11-16 October 1986 CE,
After presentation of papers on the topic of artificial insemination (test-tube babies), and listening to the explanations of experts and doctors, and after a discussion in which it was explained to the council that the methods of artificial insemination that are known nowadays are seven, the council determined the following:
Firstly: the five following methods of artificial insemination are prohibited according to sharee‘ah and are completely disallowed, in and of themselves, and because of what they result in of mixing of lineages, confusion as to who is to be regarded as the mother, and other matters that are contrary to Islamic teaching.
The first of these (prohibited) methods is where sperm taken from the husband is used to fertilise an egg taken from a woman who is not his wife, then the embryo is implanted in the uterus of his wife. End quote.
We have limited our quotation here to the case asked about in the question. See: Qaraaraat Majma‘ al-Fiqh al-Islami (p. 74 – published by the Qatari Ministry of Awqaaf).
So it is not permissible for you to donate your eggs to your sister, but you should offer a great deal of supplication for your sister and she should also offer a great deal of supplication; she should not despair of the mercy of Allah, may He be exalted, and she should take practical measures to seek treatment, for Allah, may He be glorified and exalted, has power over all things, and He blessed His Prophet Ibraaheem with children after he grew old, as He, may He be glorified, also did for His slave Zakariyyah (peace be upon them both.
And Allah knows best.