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This belief causes millions of people to throng to temples throughout the country.
The belief is that God(specifically Lord Vishnu) does grant to those who pray to Him whatever they ask for as long as it is righteous.
However, my question is whether this belief is supported by scriptures, or teachings of acharyas. Please provide any quotes from scriptures if there are any.

EDIT: This question has been wrongly marked a duplicate of another question. This question asks if God grants us our wishes and what is the support for that in the scriptures, while the other question asks about what is his motive for granting our wishes which is not the topic of this question.

This is described in Śrīmad-Bhāgavatam (Bhāgavata Purāṇa): Skandha 10: Adhyay 88:

SB 10.88.1 — King Parīkṣit said: Those demigods, demons and humans who worship Lord Śiva, a strict renunciant, usually enjoy wealth and sense gratification, while the worshipers of the Supreme Lord Hari, the husband of the goddess of fortune, do not.

Śrī Śukadeva replied that same question was asked by Yudhiṣṭhira and Śrī Kṛṣṇa answered as below:

SB 10.88.8 — The Personality of Godhead said: If I especially favor someone, I gradually deprive him of his wealth. Then the relatives and friends of such a poverty-stricken man abandon him. In this way he suffers one distress after another.

SB 10.88.9 — When he becomes frustrated in his attempts to make money and instead befriends My devotees, I bestow My special mercy upon him.

SB 10.88.10 — A person who has thus become sober fully realizes the Absolute as the highest truth, the most subtle and perfect manifestation of spirit, the transcendental existence without end. In this way realizing that the Supreme Truth is the foundation of his own existence, he is freed from the cycle of material life.

SB 10.88.11 — Because I am difficult to worship, people generally avoid Me and instead worship other deities, who are quickly satisfied. When people receive kingly opulences from these deities, they become arrogant, intoxicated with pride and neglectful of their duties. They dare to offend even the demigods who have bestowed benedictions upon them.

So, Lord Vishnu first test his devotees properly and then grant them boons. In other hand, Lord Shiva quickly gets pleased (hence known as Ashutosh) and grant boons. The story of Vṛkāsura is mentioned in same chapter of Bhagvatam to explain this.

To fulfill all your desire he gives multiple birth..So desire is the cause of rebirth..so god gave you body to fullfill your desire. Instead of desiring material things lets us all have highest desire of bhakthi or devotion towards lord. If you are asking you ask for faith, belief and devotion towards lord, that would in turn takes us in the right direction..

Prahalad and Dhurv was asked what you want, prahald said it will business transaction that if he is asking him, finally he insist, prahald asked the desire not desire anything from god in the future.
simular Dhurv told lord vasudeva, that all his desire is fulfilled just by darshan so no more desire, but says that only lord makes the desire, Asks him to make the right desire.