Is it obligatory to fulfill a vow to fast one day for every day a family member wakes up for Fajr prayer, given the regularity of waking up on many days?

A vow conditional on the occurrence of a specific event must be fulfilled when that event takes place. Therefore, the questioner is obligated to fast one day for every day she wakes up and finds one of her family members awake. If she is unable to fast due to a valid excuse, such as an illness from which recovery is not expected, then the obligation falls away, and she must feed one poor person for each day, and offer the expiation for a broken oath.