Shulchan aruch Yoreh dea 233,1: קטן בן י"ב שנה ויום אחד וקטנה בת י"א שנה ויום אחד אם יודעים לשם מי נדרו ונשבעו נדריהם נדר ושבועתם שבועה אע"פ שלא הביאו ב' שערות ובודקין שנת י"ג לזכר ושנת י"ב לנקיבה וקודם שיש י"ב לזכר וי"א לנקיבה אפילו יודעים לשם מי נודרים אינו נדר - A minor boy from 12 and 1 day or a minor girl form 11 and 1 day as long as they know to whom they are making a vow to (to Hashem) their vows are valid and their oaths are oaths even though they haven't matured. But under 12 for a boy or 11 and bellow for a girl even if they know to whom they are vowing to their vows are null.
Shulchan aruch Yoreh dea 234,1: האב מפר נדרי בתו כל זמן שהיא קטנה או נערה אפי' נדרה על דעת רבים - The father can nullify his daughters Neder while she is a minor (from 11 - 12 as under 11 her neder is not even valid) even a vow that was done in the knowledge of many people (2 people).
There is a Machlokes rishonim which vows the father can nullify, The vows that cause her physical pain and those that affect between him and her,(rosh, ritva, Ran, nimmukei yosef and tosfos) or any neder whatsoever (Rambam hilchos Nedarim 12,1 see kesef mishna there).
But here everyone agrees that this is between him and her as he has to pay up as she doesn't own any money yet ,so as long as he is mefer (nullifies) ביום שמעו on the day he hears the vow/oath (numbers 30,6) her neder is nulified.
What happens when she goes out of her fathers reshus (Domain)?
According to Ran, Rosh, tosfos and nimukei yosef ritva then the neder comes back on to her and she has to now fulfill it . this is because SA YD, 234,55: ודברים שבינו לבינה אין ההתרה אלא לעצמו כל זמן שיש לו בו תועלת דהיינו כל זמן שהיא תחתיו - Matters between her and him there is nullification of the vow only when he has a reason i.e it affects him. That is only whilst she is in his domain.
However according to the Rambam the vow is completely nullified as the father has complete power over whatever neder she says even if it doesn't affect him whatsoever.