Answered by Shaykh Abdul-Karim Yahya, SunniPath Academy Teacher
In the Shafi’i School can we make any supplication while prostrating; or is it limited to Quran and prophetic invocations? Can we supplicate at that time after saying, “Subhaana Rabbiya l-A’laa” three times?
Praise is for Allah, the Lord of all beings
May Allah bless our liegelord Muhammad, his folk and companions and grant them all peace
One may certainly supplicate while prostrating. Shaykh Ibn Hajar (Allah have mercy on him) said in Tuhfa: “It is sunna to supplicate during one’s prostration, due to the tradition related by Muslim, ‘The closest a slave is to his Lord is when he is prostrating, so strive in supplication (i.e. therein);’ and prophetic invocations (ma’thur) are best.” So one may make any supplication, but those narrated from the sunna are best. This is the case for Arabic supplication. In the case of non-Arabic supplications, they are only allowed for someone who is unable to pronounce Arabic invocations and must be limited to invocations narrated from the sunna for that point in the prayer (please see link to related answer). One may also beseech Allah in one’s heart (not moving one’s tongue) in any language and Allah’s grace is vast. And Allah knows best and he alone gives tawfiq (success).
 Ibn Hajar al-Haytami, Hawashi al-Sharwani Wa Ibn Qasim al-Abbadi ‘Ala Tuhfah al-Muhtaj Bisharh al-Minhaj (Beirut, Dar Ihya’ al-Turath al-‘Arabi), 3:270-273 and 2:103.