How can a legitimate marriage contract (Fatihah recitation) be annulled with a man who abandons prayer and drinks alcohol, given the absence of official documents proving the marriage, and in the event that the husband refuses divorce? And what is the solution if the courts only accept official documents?

If the young man drinks alcohol and abandons prayer, then you erred by agreeing to marry him. Prayer is the greatest matter in religion after faith. Abandoning it out of denial expels one from the faith, and abandoning it out of laziness is considered by some to be an expulsion from faith. Alcohol is the mother of evils, and drinking it is one of the greatest major sins. Since the legal marriage contract has been concluded, you cannot separate from him except through divorce, khul‘, or a judge's ruling. Khul‘ can occur without the husband's consent by a ruling from two arbiters or the Muslim community. According to those who consider abandoning prayer to be major disbelief (kufr akbar), the marriage contract is annulled, and the annulment of the contract depends on a ruling of the husband's disbelief. If you are unable to take the matter to court due to the contract not being documented, then consult trustworthy scholars in your country.