How can one marry a girl who has no guardian (wali) and cannot come to the mosque, and is it sufficient to bring a paper from her stating her desire for marriage for the imam to solemnize her marriage contract, even if he has not seen her before?

If a girl has no guardian (wali), the Sharia judge marries her. If there is no judge, a man of standing marries her. If there is no such man, then a Muslim man marries her. The validity of the marriage contract requires the presence of two Muslim witnesses. If the girl is in one country and the husband in another, and she has no guardian, her guardian can be someone from her city (like the mosque imam). In this case, someone can be appointed as an agent on his behalf in the husband's city to perform the contract. Alternatively, the contract can be performed between the husband and the girl's guardian via phone or internet, with two witnesses present who hear the offer and acceptance. Sheikh Ibn Baz issued a fatwa permitting marriage contracts via phone if they are secure from manipulation and all legitimate contractual conditions are met.