Will one be rewarded for having intended to observe a supererogatory fast but was prevented from doing so?

If by "family" the parents are meant, then obedience to them takes precedence over supererogatory fasting. One who is prevented by their parents from observing a supererogatory fast should break it, and they will still be rewarded for the fasting. Obey your parents from the outset, without needing food to be placed before you.