Is it permissible to give an amount vowed (200 US dollars for those afflicted by the earthquake in Pakistan) to the local mosque for its disposal, or should it be distributed to the poor and needy, in the event that it is not possible to deliver it directly to those afflicted?

Your statement, "If this problem passes peacefully, I will vow to the Most Merciful (Allah) $200," is a vow (nadhr), and it is recommended for you to fulfill it yourself or by appointing someone to deliver the money to Pakistan. If you are unable to do so, it suffices for you to spend it on the Muslim poor and needy in your own country.