Is it permissible to pay the cost of painting the donation box in cash to the person in charge of the mosque's services, or to deposit the amount in the box, instead of painting it after vowing to do so?

If painting the box does not benefit it, then your vow is from the permissible vows, and the more preponderant opinion is that it is not binding, and nothing is required of you.

However, if painting the box benefits it, then your vow is a vow of obedience (qurbah) that must be fulfilled. If you are unable to fulfill it due to circumstances beyond your control, then you must offer an expiation for a broken oath (kaffarat yameen).