Is it permissible to vow to fast for a day as a thank you to Allah after a need has been fulfilled, and to offer it as charity with the intention of having a supplication answered and a need fulfilled? Or must one fast it before its fulfillment to be considered charity?

A vow (nadhr) is a commitment made by a legally responsible Muslim to perform an act of obedience that was not obligatory before the vow, even if it is contingent upon something happening. You may vow to fast certain days if your wish is granted. If it is granted, then you are obliged to fulfill your vow. If your wish is not granted, then there is nothing upon you. It is better to supplicate frequently and draw near to Allah through supererogatory acts of worship, because a contingent vow is disliked by some scholars as it is seen as extracting from a miserly person.