Has it been reported from the Prophet (peace be upon him) a prayer known as Salatul Taysir (Prayer of Facilitation), performed in four rak'ahs, in which specific invocations are recited 100 times in each rak'ah after Al-Fatiha, as mentioned in the question, and is it permissible to pray it for the facilitation of matters?

Drawing closer to Allah through prayer and supplication helps a Muslim fulfill their needs. Allah has commanded us to seek help through patience and prayer: "And seek help through patience and prayer" (Quran 2:45), and "Or who responds to the distressed when he invokes Him?" (Quran 27:62), and "And your Lord says, 'Call upon Me; I will respond to you'" (Quran 40:60).

The Prophet (peace be upon him) would resort to prayer when facing important matters, but a prayer specifically named "Taysir" (facilitation) with the described method has not been mentioned, and acts of worship are based on divine revelation.

What has been reported is what scholars have called "Salat al-Hajah" (the prayer of need), which consists of two rak'ahs, followed by praising Allah and sending blessings upon the Prophet (peace be upon him), and then supplicating with the reported supplication: "There is no god but Allah, the Forbearing, the Generous..."

This hadith has a weak chain of transmission, but many scholars have acted upon it because it concerns virtuous deeds and has supporting evidence. The majority of scholars hold that it consists of two rak'ahs, while the Hanafis hold that it consists of four rak'ahs.