Praise be to Allah.
It is not permissible for a man who is in ihraam to wear socks, but it is permissible for a woman to do that.
Shaykh Ibn ‘Uthaymeen said in al-Sharh al-Mumti’ (7/154): Question: is it haraam for her (i.e., the woman in ihraam) to wear socks?
The answer is no. Socks are haraam for men (in ihraam).
If a man in ihraam needs to wear socks, it is permissible for him to wear them, but he has to offer the fidyah (ransom), which is either to sacrifice a sheep, or to feed six poor persons, or to fast for three days; he may do whichever he wants of these three.
The Standing Committee was asked: what is the ruling on wearing socks on the feet and doing tawaaf al-qudoom in them during Hajj, and doing the tawaaf of ‘Umrah in them during ‘Umrah?
It is not permissible for men to wear socks when in ihraam for Hajj or ‘Umrah. If he needs to wear them because of sickness etc, then he must offer a ransom (fidyah), which is to fast for three days, or to feed six poor persons, giving each one half a saa’ of dates or similar food, or to sacrifice a sheep.
Fataawa al-Lajnah al-Daa’imah li’l-Buhooth al-‘Ilmiyyah wa’l-Ifta’, 11/183.