Is expiation for a vow (kaffarat al-nadhr) obligatory for someone who vowed to place a specific amount of money in the mosque on the same day a certain matter is fulfilled, and Allah indeed fulfilled that matter for him, but he was unable to place the money on that same day?

Undertaking a vow is disliked (makrooh). What you have undertaken is a conditional vow, which must be fulfilled if the condition is met. If you are unable to fulfill it at its appointed time, then fulfill it afterwards, and there is no blame upon you. However, if you were able to fulfill it at its appointed time but failed to do so, then you must repent to Allah Almighty, fulfill the vow, and pay an expiation for a broken oath (kaffarat yameen) due to that delay.