What is the ruling on vowing an ‘Umrah for a mother who is unable to travel on her own, and the one who made the vow is not financially capable of accompanying her? Should he give her the cost of the trip, or should he give charity on her behalf?

A vow to financially support one's mother is a vow of obedience [to Allah], and it must be fulfilled, for the Hadith states: "Whoever vows to obey Allah, let him obey Him." If the intention was to accompany her, then wait until you are able, for fulfilling an unspecified vow can be done at leisure. If the vow was to give her money for her to perform 'Umrah, and she refused, you are absolved, as you have done what was required of you.