In the Qur’an, Almighty Allah teaches us: “Ask Allah of His bounty. Lo! Allah is ever Knower of all things.” (An-Nisa’: 32) Also: “ And when My servant question thee concerning Me, then surely I am nigh. I answer the prayer of the suppliant when he crieth unto Me. So let them bear My call and let them trust in Me, in order that they may be led aright.” (Al-Baqarah: 186)
As for the supplications other than the tasbih during prostration, we can say that the Muslim is permitted to ask Allah SWT for what one needs of the good things in this world and in the Hereafter, and the best of times to do that is
when prostrating, after reciting the prescribed dhikr, because the Prophet (peace and blessings be upon him) said: “I have been forbidden to recite Qur’an when bowing and prostrating. As for ruku’ (bowing), glorify your Lord therein, and as for sujood (prostration), strive in du’a’, for then it is more likely that your Lord will answer you.” (Muslim, 738).
As for asking Allah for worldly needs during the sujud, we can say that there is nothing wrong with that as long as one doesn’t go into extremes in his supplications such as asking for the impossible or asking for severing ties of kinship as these things and their likes should be avoided.
Having said this, we would like to stress that while mentioning the etiquettes of the du`aa, Muslim scholars state that the voice of the suppliant should be neither so loud nor so low. It is also part of the Sunnah to repeat the supplications three times, if you can. If not, one time will be sufficient, in sha’ Allah.