Is it permissible for my wife to perform tayammum to pray Fajr, due to her inability to perform ghusl at night or at dawn for fear of water affecting her hair, provided that she performs ghusl during the day?

When a Muslim experiences janabah, they must perform ghusl (ritual bath) for prayer. Tayammum (dry ablution) is not permissible as a substitute for ghusl except with a legitimate Shar'i excuse, such as fear of harm from using water. The harm mentioned by the questioner (the wife being harmed if she sleeps with wet hair) is not an excuse that permits tayammum, because drying the hair after ghusl is possible. If the harm is imaginary or can be avoided by drying the hair, then she must perform ghusl with water. However, if the harm is real and she cannot avoid it, then she should wash the parts of her body that will not be harmed and perform tayammum for what she fears harm from. This is based on the Almighty's saying: "So fear Allah as much as you are able" (Quran 64:16) and "Allah does not burden a soul beyond its capacity" (Quran 2:286).