He vowed to take his mother for ‘Umrah on the day of Eid; can he take her for ‘umrah in Ramadan?

If you took your mother for ‘Umrah during Ramadan, there is
nothing wrong with that, because you have done something that is better than
the time mentioned in your vow. This is like if a person vowed to pray in
al-Masjid al-Aqsa then he prayed in al-Masjid al-Haraam or al-Masjid
al-Nabawi, because he offered the prayer in a better place. In a saheeh
hadeeth it refers to that scenario and says that there is nothing wrong with
it. It was narrated from Jaabir (may Allah be pleased with him) that a man
said on the day of the conquest of Makkah: O Messenger of Allah, I vowed
that if Allah granted you victory in Makkah, I would pray in Bayt al-Maqdis
(Jerusalem). He said: “Pray here.” He asked him, and he said: “Pray here.”
He asked him again and he said: “Then it is up to you.” Narrated by Ahmad
and Abu Dawood; classed as saheeh by al-Haakim. 

But if your mother refuses to do ‘Umrah, there is no
obligation on him either, because he has done what he had to do and the
refusal came from someone else. 

And Allah is the source of strength. May Allah send blessings
and peace upon our Prophet Muhammad and his family and Companions. End
quote. 

Standing Committee for Academic Research and Issuing Fatwas 

Shaykh ‘Abd al-‘Azeez ibn Baaz, Shaykh ‘Abd al-Razzaaq ‘Afeefi, Shaykh ‘Abd-Allaah ibn Ghadyaan