Is it permissible for someone who vowed to perform Umrah for his mother every Eid to perform it for her in Ramadan instead of Eid without expiation, and what is required of him if his mother does not wish that?

If your mother performed Umrah (definition) during Ramadan (definition), then there is nothing upon you, because you have fulfilled something better than the vowed time. This is similar to someone who vowed to pray in Al-Aqsa Mosque but prayed in the Sacred Mosque (Masjid al-Haram (definition)) or the Prophet's Mosque (Masjid an-Nabawi), as he performed it in a superior place. It has been narrated in an authentic hadith (definition) that there is nothing upon him in this matter. It is narrated from Jabir (may Allah be pleased with him) that a man said on the day of the conquest (of Mecca): "O Messenger of Allah, I vowed that if Allah grants you the conquest of Mecca, I would pray in Jerusalem (Bayt al-Maqdis)." He (the Prophet) said: "Pray here." He asked him again, and he said: "Pray here." He asked him again, and he said: "Then do as you wish." (Narrated by Ahmad and Abu Dawud, and authenticated by Al-Hakim). However, if his mother refused to perform Umrah, there is also nothing upon him, because he fulfilled his part, and the refusal came from someone else.