Is it permissible to donate the value of the udhiyah to the mosque fund, especially since the person offering the sacrifice lives in a country where there are no poor people and he has no relatives, while the mosque is in dire need of money for its expansion?

The Udhiyah (sacrificial offering) is a confirmed Sunnah, and it is better than giving its value in charity, because it is a timely act of worship. It can be performed by appointing someone to slaughter it in another country if there are no poor people [in one's own country]. Donating for the construction of a mosque does not contradict it; rather, both can be combined. Indeed, building mosques is among the greatest acts of closeness [to Allah], and spending in the way of Allah increases provision.